Copyright 19 7 9
Todd L. Walton Jr.
Lif^oiAL Bmj3 immsmm ow benches
TQDD L. WALTON JR.
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
ACKNOWLEDGEMENTS
With utmost sincerity, the author wishes to thank Dr.
B.A. Christensen and Dr. T. Y. Chiu for their advice, direc
tion, and considerable assistance which they have given to
the author's Ph.D. program. Without their untiring help and
encouragement this thesis would not have been possible.
Thanks are also due to Dr. D. P. Spangler, Dr. F. W.
Morris, and Dr. J. G. Melville, for serving as the author's
supervisory committee.
The author also wishes to express his deep appreciation
to his former mentor, Dr. R. G. Dean whose efforts in helping
the author to understand the basics of science have never
ceased, and to his good friend the late Professor J. A.
Purpura whose encouragement and personal understanding of
shore processes were of tremendous benefit to the author.
Appreciation is also extended 'to the Florida Sea Grant
Program whose grant C0M315843 ) made many of the early
results of this thesis possible, and the Department of
Natural Resources, Bureau of Beaches and Shores, State of
Florida, whose support and encouragement have benefitted the
author .
Finally, the author wishes to thank his wife Cindy
whose patience and unselfishness throughout the pursuit of
this dissertation has made life more wonderful.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iii
LIST OF TABLES vii
LIST OF FIGURES ix
LIST OF SYMBOLS xvi
ABSTRACT xxiv
CHAPTER
I. INTRODUCTION 1
11 Introductory Note 1
1.2 Statement of Erosion Problem 3
1.3 Objectives 17
II. REVIEW OF SURF ZONE MODELS 19
2.1 Review of Existing Longshore Sand
Transport Models 19
2 . 2 Review of Existing Longshore
Current Models 2 7
2.3 Review of Friction Factor 39
2.3.1 Field Data i+0
2.3.2 Laboratory Data i44
III. REVIEW OF EXISTING DATA ON LONGSHORE
SAND TRANSPORT 51
3.1 Introductory Remarks 51
3.2 Laboratory Studies 51
3.2.1 Krumbein Laboratory Study 53
3.2.2 Seville Laboratory Study 5 6
3.3.3 Shay and Johnson Laboratory Study . 56
3 . 3 . M Sauvage and Vincent Laboratory
Study 5 8
3.3.5 Savage and Fairchild Laboratory
Study 5 8
3.3.6 Price, Tomlinson, and Willis
Laboratory Study 6 0
3.2.7 Barcelo Laboratory Study 61
3.2.8 Bijker Laboratory Study 61
IV
Page
3.3 Field Studies 62
3.3.1 Watt Field Study 5 5
3.3.2 Caldwell Field Study 69
3.3.3 CERC Field Study 70
3.3.4 Moore and Cole Field Study .... 70
3.3.5 Komar Field Study 71
3.3.6 Thornton Study 7 2
IV. SAND TRANSPORT MODEL DEVELOPMENT 7 3
4.1 Introduction 7 3
M . 2 Review of Mass, Momentum, and Energy
Conservation in Periodic Wave Flow ... 76
4.3 Review of Wave Field Equations 87
4.14 Proposed Sand Transport Model 105
4.4.1 Longshore Current Estimation . , .10 5
4.4.2 Integrated Sand Transport
Estimation 120
4.4.3 Sand Transport Distribution
Across Surf Zone 131
4.4.4 Estimation of Efficiency
Factors and Dynamic Friction
Coefficient 13 8
4.4.5 Model Comparison with Laboratory
and Field Results 155
V. APPLICATION OF SAND TRANSPORT EQUATION
TO CALCULATION OF LITTORAL DRIFT
USING SHIP WAVE DATA 16 2
5.1 Application of Model 16 2
5.1.1 Data Source' 16 4
5.1.2 Analysis of Wave Data to
Compute Sand Transport 16 5
5.2 Results of Sand Transport
Computations 17 2
5.2.1 Use of a Littoral Drift Rose . . . 175
5.2.2 Possible Sources of Data
Error or Bias 178
5.2.3 Other Possible Errors or
Bias in Analysis 180
5.2.4 Other Potential Sources
of Error 182
5 . 3 Comparison of Calculated Littoral
Drift Rates with Previously
Estimated Values 185
5.4 Comparison of Estimated and
Observed Wave Climates 19 8
VI. ANALYTICAL MODELS FOR SHORELINE CHANGE . . . 20 8
6.1 Introductory Rem^arks 208
V
Page
6 . 2 Heuristic "Equilibrium"
Shoreline Model Development 209
6 . 3 , Analytical Treatment of Shoreline
Change Model 229
VII. SAND TRANSPORT AND STORAGE AT INLETS .... 248
VIII. SUMMARY AND CONCLUSIONS 259
REFERENCES 262
APPENDIX
A. DERIVATION OF LONGSHORE ENERGY FLUX
PER UNIT LENGTH SHORELINE 275
B. LONGSHORE SAND TRANSPORT MODEL DATA
SUSPENDED SAND CONCENTRATION DATA 2 80
C. ANALYSIS OF SSMO WAVE HEIGHT, PERIOD,
AND DIRECTION RANGES 289
BIOGRAPHICAL SKETCH . ' ■ . 321
vi
LIST OF TABLES
Table Page
1.1 Volumes of Material Present in Outer
Inlet Shoals of Florida Inlets . 11
1.2 Beach Nourishment Projects 12
2.1 Littoral  Sand Transport, .Models 20
2.2 Littoral Sand Transport Models of
Equation Type 2.1 23
2.3 Longshore Current Formulas 29
2.4 Results of Field Measurements of Fric
tion Factor i43
2.5 Results of Laboratory Measurements of
Friction Factor i+8
3.1 Testing Variables of Laboratory Data 54
3.2 Summary of Wave Basin Dimensions 55
3 3 Field Study Tests for Sand Transport
Determination 56
4 .1 Observed and Theoretical Values of
4.2 Integrals for Sediment Transport 137
5.1 Comparison of Annual Net Littoral Drift
Rates as Estimated by the U.S. Army Corps
of Engineers and as Calculated in the
Present Study 183
5.2 Comparison of Annual Average Total Lit
toral Drift Rates as Estimated by the
U.S. Army Corps of Engineers and as
Calculated in the Present Study 187
vi.i
LIST OF TABLES
(Continued)
Table Page
5.3 Recording Periods of Shore Based CERC
Wave Gages Used in Comparison of Actual
to Predicted Shore Wave Climate 200
B.l Shay and Johnson Data 280
B.2 Komar Data 281
B.3 Fairchild Data 282
B.4 Watts Data 283
B.5 Kana Data 2B^■
B. 6 Fairchild Data 285
C. l Representative Values of Wave Height Used
in Computation of Longshore Energy Flux 291
C.2 Representative Values of Wave Period Used
in Computation of Longshore Energy Flux 292
::v ill
LIST OF FIGURES
^^g^^ Page
1.1 Erosion Situation in Florida 2
1.2 Relative Rise in Sea Level in Florida t
1.3 Bruun's Concept of Beach Profile
Response to Sea Level Rise 5
1.4 Sediments Deposited off Central Padre
Island During and After Hurricane Carla
on a Mud Bottom 7
1.5 General BeachDune Profile Before and
After Hurricane Eloise Between Panama
City and Destin, Florida 9
1.5 Deposition of Sand in the Interior of
St. Lucie Inlet 14.
1.7 Relationships Between Outer Bar Stor
age and Cross Sectional Area of Inlets 15
3.1 Distribution of Longshore Velocity and
Sediment Transport Ac:toss the Surf Zone , 61+
14.1 Shear Stresses Acting on the Faces of
a Water Column 84
4.2 Schematic of the Sur;' Zone 92
4.3 Schematic Plan and Section of the Near
shore Region 108
4.4 Shear Forces Due to Oscillatory Wave
Velocity and Longshore Current 115
4.5 Relation of Normal Force to Moving Force ...... 123
4.5 Plot of Sand Distr'bution Across the
Surf Zone 135
4.7 Sediment Suspension Due to Cb) Long
Periods, High Fall Velocities, (c) Short
Periods, Low Fall Velocities li+O
LIST OF FIGURES
( Continued )
Figure Page
4.8 Vertical Forces on a Grain Particle
About to be Lifted from the Bed 144
4.9 Relationship Between Sediment Mixing
Coefficient and Mixing Parameter — g — ~ 150
4.10 Relationship Between Sediment Concentra
tion and Lift to Weight Ratio Parameter 151
4.11 Postulated Values of Efficiency
Factor e, 154
a
4.12 Variation of Dynamic Friction Angle
with Solids Reynold's Number 155
4.13 Sand Transport Model Relationship 158
4.14 Best Fit Sand Transport Model for
Fairchild Data 159
4.15 Best Fit Sand Transport Model for
Field Data 160
5.1 Definition of Azimuth Angle Normal to
Shore 9 , and Azimuth Angle of Wave
Propagation 9 168
5.2 Relationship Between Direction of Wave
Propagation and Direction of Longshore
Sand Transport 171
5.3 Location of SSMO Data Squares Adjacent
to the Florida Peninsula 173
5.4 Azimuth of Normal to Shoreline at Ponte
Vedra Beach, Florida 175
5.5 Determination of Total Positive and Total
Negative Littoral Sand Transport 177
5.6 Ideal Case of an Unstable Null Point 191
5.7 Ideal Case of a Stable Null Point 193
LIST OF FIGURES
(Continued)
FigTjre . Page
5.8 Stable Type Littoral Drift Rose Due to
Unimodal Wave Climate 195
5.9 Unstable Type Littoral Drift Rose Due to
Bimodal Type Wave Climate 196
5.10 Instability Formed Capes in Santa Rosa
Sound 197
5.11 Comparison of Computed and Observed Wave
Heights at Daytona Beach, Florida 201
5.12 Comparison of Computed and Observed Wave
Periods at Dajrtona Beach, Florida 202
5.13 Comparison of Computed and Observed Wave
Heights at Lake WorthPalm Beach, Florida 203
5.14 Comparison of Computed and Observed Wave
Periods at Lake WorthPalm Beach, Florida 2Q^■
5.15 Comparison of Computed and Observed Wave
Heights at Naples, Florida 205
5.16 Comparison of Computed and Observed Wave
Periods at Naples, Florida 205
6.1 Logspiral Curve Fit to Shoreline at
Riomar, Florida 211«
6.2 Logspiral Curve Fit to Shoreline at
Lacosta Island 212
6.3 Logspiral Curve Fit to Panhandle Coast . 213
6.1+ Orientation of a Shoreline Segment Having
Equal Offshore Wave Energy 217
6.5 Wave Energy Rose for Ocean Data Square
#12 off East Coast of Florida 219
6.6 Energy Diagram of Coastline with Equilib
rium Coast Shape 221!
LIST OF FIGURES
(Continued )
Fig^e Page
6.7 Solution Curves to Equilibrium Shoreline
Equation for the East Coast of Florida 222
5.8 Equilibrium Shoreline Shape for Riomar,
Florida 22l
6.9 Solution Curves to Equilibrium Shoreline
for Gulf Coast of Florida 226
5.10 Equilibrium Shoreline Shape for Lacosta
Island, Florida 22?
5.11 Equilibrium Shoreline Shape for Panhandle
Coast of Florida 228
6.12 Equilibrium Shoreline Orientation for
Shoreline Protected by Offshore Break
waters on the East Coast of Florida 230
6.13 Definition of Sign Convention for Shore
line Modeling 233
6.14 Solution Curve for Case of Sand Buildup
at a Structure 239
6.15 Solution Curve for Sand Buildup at Struc
ture (After Time Sand has Reached End of
Structure ) 243}
6.16 Beach Nourishment Plan Shapes for Beach
Fill 242
5.17 Beach Nourishment Plan Slopes for Triangu
lar Initial Distribution 2m
6.18 Beach Noiarishment Plan Shapes for Gap in
Beach Fill , 24S
6.19 Solution Graph for One End of Semiinfinite
Beach Nourishment Project 245
7.1 Tidal Pr ism— Outer Bar Storage Relationship
for Highly Exposed Coasts 25S
xii.
Copyright 19 7 9
by
Todd L. Walton Jr,
LIST OF FIGURES
(Continued)
Figiire Page
7.2 Tidal PrismOuter Bar Storage Relationship
for Inlets on Sandy Coasts 25U
7.3 Tidal PrismOuter Bar Storage Relationship
for Moderately Exposed Coasts 253'
7.1+ Tidal PrismOuter Bar Storage Relationship
for Mildly Exposed Coasts 255
A.l Definitions for Conservairion of Energy
Flux for Shoaling Wave 276
C.l Variation of Average Annual Total Littoral
Drift with Beach Orientation — Fernandina
Beach to St, John's River, Florida 295
C.2 Variation of Average Annual Total Littoral
Drift with Beach Orientation — St. John's
River to St. Augustine Inlet, Florida 296
C.3 Variation of Average Annual Total Littoral
Drift with Beach Orientation — St. Augus
tine Inlet to Ponce de Leon Inlet, Florida 297
C.^■ Variation of Average Annual Total Littoral
Drift with Beach Orientation — Ponce de
Leon Inlet to Cape Kennedy, Florida 298
C.5 Variation of Average Annual Total Littoral
Drift with Beach Orientation — Cape Kennedy
to Sebastian Inlet, Florida 299
C.6 Variation of Average Annual Total Littoral
Drift with Beach Orientation — Sebastian
Inlet to Fort Pierce Inlet, Florida 300
C.7 Variation of Average Annual Total Littoral
Drift with Beach Orientation — Fort Pierce
Inlet to St, Lucie Inlet, Florida 301
C.8 Variation of Average Annual Total Littoral
Drift with Beach Orientation — St. Lucie
Inlet to Jupiter Inlet, Florida 302
xxii
LIST OF FIGURES
(Continued)
Figure Pag^
C.9 Variation of Average Annual Total Littoral
Drift with Beach Orientation — Jupiter Inlet
to Lake Worth Inlet, Florida 303
C.IO Variation of Average Annual Total Littoral
Drift with Beach Orientation — Lake Worth
Inlet to Hillsboro Inlet, Florida 304
C.ll Variation of Average Annual Total Littoral
Drift with Beach Orientation — ^JiilJ_sboro
Inlet to Cape Florida, Florida 305
C.12 Variation of Average Annual Total Littoral
Drift with Beach Orientation — Perdido Pass
to Pensacola Bay Entrance, Florida 306
C.13 Variation of Average Annual Total Littoral
Drift with Beach Orientation — Pensacola
Bay Entrance to Choctawhatchee Bay
Entrance, Florida 307
C.14 Variation of Average Annual Total Littoral
Drift with Beach Orientation — Chocta
whatchee Bay Entrance to St. Andrew Bay
Entrance, Florida 308
C.15 Variation of Average Annual Total Littoral
Drift with Beach Orientation — St. Andrew
Bay Entrance to St. Joseph Bay Entrance,
Florida 309
C.16 Variation of Average Annual Total Littoral
Drift with Beach Orientation — St. Joseph
Bay Entrance to Cape San Bias, Florida 310
C.17 Variation of Average Annual Total Littoral
Drift with Beach Orientation — Cape San
Bias to Cape St. George, Florida 311
C. 18 Variation of Average Annual Total Littoral
Drift with Beach Orientation — Anclote
Keys to Clearwater Pass, Florida 312
C.19 Variation of Average Annual Total Littoral
Drive with Beach Orientation — Clearwater
Pass to Tampa Bay Entrance, Florida 313
xiv
LIST OF FIGURES
(Continued)
^ig^e Page
C.20 Variation of Average Annual Total Littoral
Drift with Beach Orientation — Tampa Bay
Entrance to Big Sarasota Pass, Florida 314
C,21 Variation of Average Annual Total Littoral
Drift with Beach Orientation Big Sara
sota Pass to Venice Inlet, Florida 315
C.22 Variation of Average Annual Total Littoral
Drift with Beach Orientation — Venice Inlet
to Boca Grande Inlet, Florida 315
C.23(a) Variation of Average Annual Total Littoral
Drift with Beach Orientation — Boca Grande
Inlet to San Carlos Bay, Florida 317
C. 23(b) Variation of Average Annual Total Littoral
Drift with Beach Orientation — Boca Grand
Inlet to San Carlos Bay, Florida 318
C.21+ Variation of Average Annual Total Littoral
Drift with Beach Orientation — San Carlos
Bay to Wiggins Pass, Florida 319
C.25 Variation of Average Annual Total Littoral
Drift with Beach Orientation — Wiggins Pass
to Cape Romano, Florida 320
C. 26 Modification of Wave Data for Waves Paral
lel to Coastline 293
LIST OF SYMBOLS
A = mixing parameter constant
= inlet cross sectional area at throat of tidal inlet
a = wave amplitude
B = proportionality factor associated with bed load
transport
Bg = proportionality factor associated with bed load
transport inside the surf zone
B^ = constants of longshore velocity distribution
^i' ^i ~ functions of bed load portion of sand transport
equation.
c = wave celerity
c^ = wave celerity at wave breaking
Cg = speed of wave energy propagation
c^ = wave celerity in deep water
C = concentration in dry weight of sediment per unit
weight fluid
= drag coefficient
= friction coefficient for flat plates
Cj^ = life coefficient
= Chezy friction coefficient
d = mean sand grain size
dgp = median sand grain size
D = n + H, total depth of water
XV i
= total depth of water at location of breaking waves
= depth of water at limits of longshore sand movement
e^ = efficiency coefficient associated with bed load
transport
e^ = efficiency coefficient associated with suspended
load transport
E = energy density
E* = quantity proportional to energy flux density
E^ = energy density in deep water
f = friction factor
f = frequency of occurrence
f^ = friction factor defined by energy dissipation for
waves
f^ = friction factor defined for oscillatory flow
= functions of sand transport equation
F = energy density flux
F . = horizontal force per unit area due to slope of
'^'^ free water surface
g = acceleration due to gravity
G = gravity force on a spherical sediment particle
h = depth below still water
hj^ = depth below still water level at wave breaking
h^^ = height of beach berm above still water level
h^ = depth below still water level in deep water
h^ = total depth of active beach profile (in longshore
sand transport)
H = wave height
H, = wave height at wave breaking
xvii
■ wave height in deep water
■ significant wave height
index corresponding . to horizontal coordinate in
the xdi^^^ection
inmersed weight sediment transport rate per unit
width
immersed weight sediment transport rate per unit
width (bed load)
immersed weight sediment transport rate per unit
width (suspended load)
total immersed weight sediment transport rate for
surf zone
total immersed weight sediment transport rate for
surf zone (bed load)
total immersed weight sediment transport rate for
surf zone (suspended load)
index corresponding to horizontal coordinate in
the ydirection
wave number
ratio between, setup slope and beach slope
constants
friction coefficient
refraction coefficient
shoaling coefficient
rip current spacing
wave length
lift force on a spherical sediment particle
wave length in deep water
mass of bed load sediments
X viii
m = mass of suspended sediments
M = summation constant
= + M^, total mass transport per unit width or
total mean momentum per unit area
= mass transport per unit width associated with
fluctuating motion
n = transmission coefficient
N = dimensionless lateral mixing coefficient
N* = normal directed, component of energy density at
shoreline
p = pressure
p = dimensionless lateral mixing coefficient
p^ = porosity
= power/unit area available for bed sediment
transport
= "longshore" energy flux
Pg = power/unit area available for suspended sediment
transport
P* = parallel directed component of energy density at
shoreline
Qi^ = total volumetric longshore sand transport rate
r = roughness parameter
r^ = ripple parameter
= total average resistance force
R = ripple height
R = equivalent of Reynolds Number in solids mixture
s
s = fraction of incoming energy flux available to the
longshore current
xix
proportionality factor associated with suspended
load transport
specific gravity of sediment
proportionality factor associated with suspended
load transport inside the surf zone
excess momentum flux tensor (radiation stress
tensor)
function of suspended load portion of sand trans
port equation
time
wave period
gradient of "radiation" stress due to periodic
wave motion
water particle velocity maximum due to wave motion
at the bed
total water particle velocity
fluctuating water particle velocity component
water particle velocity due to wave motion
mean velocity, component normal to the beach
mean velocity of bed load transport in the long
shore direction
mean velocity of suspended load transport in the
longshore direction
+ , total mean transport velocity
mean transport velocity associated with mean
motion
mean transport velocity associated with fluctu
ating motion
resultant velocity vector of combined wave and
current motion
velocity component parallel to the beach
xx:
dimensionless velocity parallel to the beach
velocity component parallel to the beach at loca
tion of breaking waves
mean velocity (across surf zone) parallel to beac:
vertical velocity component
fall velocity of sand grain
horizontal coordinate perpendicular to the beach
dimensionless distance from the shoreline
width of the surf zone
horizontal coordinate parallel to the beach
vertical coordinate
incident wave angle
incident wave angle at breaking
incident wave angle at wave generator
incident wave angle in deep water
bottom slope
specific weight of fluid
dimensionless suspension height
kronecker delta
energy dissipation function
energy dissipation due to bottom friction
vertical sediment mixing coefficient
surf similarity parameter
water surface elevation
mean water surface elevation
azumth angle of wave ray
XX i
®n ~ outward directed normal of shoreline
ic = ratio between breaking wave height and the depth
of water at breaking
IC" = shoreline diffusivity value
X = phase function
A = linear spacial concentration in sedimentfluid
mixture
i = dynamic viscosity of fluid
^ei ~ dynamic eddy viscosity of fluid
V = kinematic viscosity of fluid
C = roughness parameter
= water particle excursion due to wave motion at bed
P = density of fluid
Pg = density of sediments
a = local wave frequency
^ij ~ shear stress
'^hi " ^^^^^ stress at the bed
(j) = dynamic friction angle
*^'s ~ static friction angle = angle of repose for sediment
$ = wave sheltering angle
X = dimensionless coefficient dependent on wave and
sediment parameters
Y = velocity potential
u = wave frequency
= resultant angle of energy density at sheltered
shoreline point
xxii
ub scripts
b = breaker line or bed load
h = bottom
m = maximum (time wise)
o = deep water
s = suspended sediment
w = wave
xxiii
Abstract of Dissertation.Er>e^ exited to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
LITTORAL SAND TRANSPORT ON BEACHES
By
Todd L. Walton, Jr.
June 19 79
Chairman: Dr. B. A. Chris tensen
CoChairman: Dr. T. Y. Chiu
Major Department: Civil Engineering
The transport of sand along beaches within the surf
zone is considered in this dissertation. The driving forces
for sand movement are considered to be the breaking water
waves which transfer a part of their energy to the shoreline
made up of noncohesive sandy material.
The primary goal of this dissertation is to develop a
model of sand transport using the excess momentum flux of a
water waves approach as a driving force for the water cur
rents and an energistics approach to develop the sand trans
port created by the water currents. Both bed load and
suspended load are considered in the model which is an
outgrowth of the past work of M. S. LonguetHiggins on
"radiation" stresses within the surf zone and of R. A.
Bagnold whose energistics approach to riverine sand transport
is adhered to in the development of a similar model for
littoral sand transport within the surf zone of sandy beaches.
The equation for littoral sand transport developed in
this dissertation has a linear (in breaking wave angle)
xxiv
component' of suspeTid^d sediment tiraTisport , and a nonlinear
(in breaking wave angle) component of bed load sediment
transport. By using published experimental data, two unknown
coefficients for the model are found. Existing data used in
model correlation indicate that the model is reasonably
valid.
As a secondary goal of this dissertation, the model is
used to develop sand t^^ansport values along the sandy beach
portions of Florida. The computed values of sand transport
are compared to existing estimates with reasonable results.
Additionally, a new method of presentation of littoral
sand transport computation results is suggested which pro
vides a means of evaluating stability/ instability features
of shorelines. An example is given of how bimodel wave
climates of elongated bays may lead to the numerous insta
bility features (giant cusps) seen in these bays.
Additional analytical modeling is proposed using linear
formulation (in wave angle) of the sand transport equation
and is found to be useful in predicting the shoreline shapes
of many coastal features.
XXV
CHAPTER I
INTRODUCTION
1 . 1 Introductory Note
Sandy Beaches have long enjoyed great popularity with
both the residents of coastal areas and the enormous influx
of tourists seeking out these vacation meccas . It has long
been recognized that these same beaches, an increasing
source of income for coastal zone citizens, are in serious
trouble due to erosion. Preservation of these beaches is
not only desirable aesthetically, but is also an economic
necessity.
The economic significance of the erosion problem can be
seen (at least in one state) from costs shared between the
State of Florida and the Federal Government in order to
preserve Florida beaches. As of 1970, estimated first costs
of authorized Federal beach improvment projects in Florida
amounted to over 76 million dollars for 108 miles of beach.
Estimated first costs to correct all the existing erosion
problems in Florida (includes authorized and unauthorized
projects) amounted to over 113 million dollars for 2 09 miles
of ocean shoreline (1).
Of approximately 1000 miles of sandy beaches in Florida
(see Figure 1.1), the annual quantity of erosion in the
1
2
GEOR Gl A
C Ferrcndina
. ; M^^st. Marys River
St. Aview's
Inlet
JockscvTville
~' John's River
St. Ajgustine Inlet
Matanzas Inlet
Daytona
pQuca Da Lecn Inist
Cops Curmeral
LEGEND
ANMJAL SHCRajNE RFCE3SCN
~~ 1 20 Ft. Plus
iO20R.
5)0Ft
SH8 25 Ff.
« I3F1
I Ql Ft.
C5
Q
Sedcsfian Inlet
Ff. Fieme Inlet
St. Luds Iniei
Jupitar Island
J'jpifBr Inlst
rbi.71 53ccij Inist
■3cynk>n iniet
3cca .Raicn Inlet
P^. E'.'iinjkzlss In (at
BalKSrs Hciiiovfr Inlet
Micmi
0
Scale
20 50 Milfl
.J i_
Key Vtet^
0 20 60 KiiciT^tars
Figure 1.1. Erosion Situation in Florida
3
nearshore area has been estimated at 15,000,000 cu. yds. per
year, with over 20% of the beach shoreline in a critical
state of erosion. Factors that influence beach erosion are
LITTORAL SAND TRANSPORT ON BEACHES
By
TODD L. WALTON JR.
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
1+
1910 !92D 1930 1940 1950 I960 1970
TIME (Yecrs)
Figure 1.2. Relative Rise in Sea Level In Florida (from Hicks ' (.2))
5
Dunea
Beach
Sea Level Aft a* Rise
Inttid .Sear Level
Initial Bottom Profile [
Hypothetical Bottom Profile After Rise in Seo Lave!
Actual Bottom Profile After
Rise Of Sea Lav^ And The
C^jcaTtftafivft Balance Bet»*eeR
Erosion And Bottom Deposits
After Sea Level Rlsa
Limiting Depth Between Predominant
Neorsftoref And Offshore Material Littoral
Drift Characteristics
Figure 1.3. Bruun's Concept of Beacli Profile Response
to Sea Level Rise (from Bruun, P. (3))
6
volume of sand from the. .upper, heaxih profile will slough off
to maintain an equilibrium bottom profile offshore. This
hypothesis was first discussed by Bruun (3) who developed a
quantitative relationship for the rate of shoreline retreat
in terms of the rate of sea level rise which, for Florida,
amounts to about 1 to 3 feet of beach recession per year.
As many areas of the coast appear to be relatively stable
(i.e. much less than 1 foot erosion per year in the short
term), it is reasonable to postulate that either the hypoth
esis is wrong or that a trend of shoreline erosion in response
to rising sea level is not gradual but rather takes place
during more severe wave activity such as occurs during
hurricanes or extratropical storms. In areas of normally
low wave energy, rapid response to high wave events does
occur and lends credulity to the above theory. Severe
erosion occurs on our shorelines during storms and complete
recovery of the lost sand is never made to the beaches. An
example of what can happen during a hurricane is shown in
Figure l,i+ from the work of Hayes (U). This figure docu
ments the potential for a large storm (Hurricane Carla) to
transport tremendous quantities of material offshore out of
the zone of all but extreme wave activity where it will not
be returned to the beaches. In this storm, over 100 feet of
dune system was virtually destroyed and a sand layer varying
in thickness from 1 to 9 cm was deposited out to depths of
7
20 Ft.
PostCaria Gi
Bed Thickness
l>9cm
I— S9cm
I— 3 6cm
1 I 3cm
m  Sand Layer
(PostCarla)
2^Ma\or Hurricane
■~" Channel
GULF OF MEXICO
Artifical
Channel
— Hypothetical
Density Current
Flow Pattern
Contour Interval
0 5
20 R.
Figure 1.4.
Sediments Deposited off Central Padre Island
During and After Hurricane Carla on
a Mud Bottom (from Hayes (4))
8
120 feet by a combination of hurricane waves and currents.
Figure 1.5 presents the results of a more recent study of
the beach erosion occurring during Hurricane Eloise which
made landfall just west of Panama City, Florida in September
1975 (5). The dune erosion profile shown is the result of a
composite profile taken over a 20mile stretch of beach from
Panama City Beach to Destin, Florida. Over 7 5 feet of large
15 to 20foot primary dune system was virtually destroyed by
this storm. It will take years before part of the dune
system will be rebuilt naturally from the sand deposited
offshore during this storm, and from experience gained from
other hurricanes the beachdune system will never recover
completely.
Sea level rise coupled with severe wave events is not
the only reason for our erosion problem though. Another
major contribution to erosion is our inlet systems and their
corresponding navigation channels either natural or artifi
cially cut through the littoral zone. In the sandy beach
littoral zone of Florida alone there are 5 7 inlets. Fourteen
of these inlets have Federally authorized navigation projects
with authorized channel depths extending to the ocean or
gulf of over 20 feet. There are at least 14 more navigation
projects with authorized channel depths of 10 feet or more.
For comparative purposes, natural controlling depths on the
outer bars of "unimproved" inlets are on the order of 6 to 8
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feet. Many of our Feder^al. navigafion projects are not
natural channels but have been cut directly through the
corresponding barrier islands such as St, Lucie Inlet on the
lower east coast of Florida. The effects of this inlet will
be mentioned later. When a channel is either cut through a
barrier island or dredged below the natural existing depths,
the flow of water through the channel to the bay (or lagoon)
on flood tide and to the oceanCor gulf) on ebb tide is
increased leading to an increased capability of the channel
to flush sand to its inner bay system or outer shoal system.
The channel also acts as a barrier to littoral sand moving
along the coast which eventually works its way into the
channel and then migrates to either the inner or outer
shoals of the inlet. The sand in the interior of the bay
systems cannot work its way back out as there is no wave
activity to agitate the sediments into suspension such that
the water can carry it out. Thus, the bay shoals of these
inlet systems act as net sinks to the beach sand system.
Volumes of material residing in the inner and outer shoal
systems have been shown to be substantial (6). An example
of the amount of sand stored in Florida's outer shoal systems
is shown in Table 1.1. In Florida, considering present
erosion rates, it has been estimated that over 200 years
worth of sand resides in the outer shoals of these inlets
(6). The sand moves to these inlets by means of littoral
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sand transport along the beaches ; the primary driving force
being wave activity. Figure l.S shows an example of inner
shoal sand trapping for St. Lucie Inlet which was cut in
1892 through the barrier island. As noted by the solid line
in Figure 1.6, the total sand deposited in the inlet over
the years 1892 to 1930 amounts to over 9 million cubic yards
of sand.
It appears £rom the trexid shown in this figure that
inlets shoal rapidly in their early years and eventually
reach an equilibrium inner shoal area (7). The trend for
outer bar shoals of inlets is not so apparent. Figure 1.7
shows a relationship proposed by Walton (8, 9) between the
volume of sand stored in the outer shoal of an inlet and the
size of the inlet as determined by its cross sectional
throat area (which is related in turn to the volume of water
flowing through the inlet). Three curves are given express
ing various severity of wave activity at the inlet. The
following sand storage equations postulated by regression
analysis by Walton (8, 9) are given for the three wave
conditions :
1 28
V = 33.1 * heavily exposed inlets
1 2 8
¥ = 40.7 A^ * moderately exposed inlets
1 2 8
V = 45.7 A^ ■ mildly exposed inlets
where ¥ = volume of sand stored in the outer bar in cubic
yards, and A^ = cross sectional area of the inlet in feet
squared .
14
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pTrend Line for Shoaling Pattern
in St. Lucie Inlet. ( From 2 miles
North of In let to I mile South of
niet in I.C.W.W.)
I — Trend Line for Shoaling Pattern
n St Lucie Inlet. (Vicinity of
3/4 mile Radius of Inlet.)
1883 18s: 1900 1910 1920 1930 1940 1950 1960
YEAR
Figure 1.5. Deposition of Sand in the Interior of St.
Lucie Inlet (from Dean and Walton (7))
15
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As seen in the . aLove. equations j the more severe the
wave activity on the outer bar, the smaller its storage
capacity is — i.e. wave activity limits the size of the outer
shoal area by driving shoal material back to the beaches.
The size of the inlet is the main controlling factor though
as larger inlets store more sand. Thus deepening an inlet
and consequent enlargement of an inlet's tidal prism may
well cause additional shoaling on the outer bar if bar
storage equilibrium has not been reached. The sand necessary
to make up this additional storage volume must come from
adjacent beaches.
Figure 1.1 is a view of the erosion situation along
Florida's coast in 196 3, but is similar to the erosion
situation today. The large black spikes where the critical
shoreline recession exists are at locations of inlets thus
confirming our expectations that inlets cause erosion to
adjacent beaches. Of course, it is also well known that in
areas of a predominant net sand transport along the beaches,
improvements such as jetties at inlets can cut off the
natural flow of sand and thereby starve downdrift beaches of
sand. In Florida, on practically every south side of an
inlet (downdrift side), erosion is excessively high. It is
not uncommon in Florida to have large stretches of shoreline
adjacent to inlets undergoing recession at rates upwards of
10 feet per year.
17
Another cause of "apparent" erosion to our shorelines
is that of barrier overwash. Many of our low barrier islands
are very susceptible to wave action occurring over the
barrier island during periods of high tides with a consequent
driving of sand into the bay systems as overwash fans. Very
little is known as to the quantities of sand .lost during
overwash events (10).
/ Causes of erosion are then due to the two modes of sand
loss; offshore sand lossybrought about by the onshore
offshore sand movement mode,^and longshore sand loss (which
moves sand toward inlet traps) brought about by the "littoral"
sand transport occurring due to wave activity on sandy
beaches. Erosion due to both modes of motion can be de
scribed by a simple continuity of sand equation and applied
to predict shoreline changes. The topic of this thesis
though will only deal with the longshore sand transport,
"littoral" drift, which appears as the major cause of erosion
when man interferes with the beach _( 112 3 ) .
1 . 3 Ob j ectives
This thesis will specifically deal with a predictive
model for determining sand transport along sandy beaches as
caused by water wave activity. Although in the vicinity of
inlets or estuaries other factors such as tidal currents
come into play, only the natural forcing function of gravity
water waves will be dealt with in this thesis. The model
18
will be applied to the jnovement .Of. cohesionless material
(sand). Using the predictive model, sand transport will
calculated for Florida's shoreline and a linearized sand
transport model will be considered for use in predictive
techniques for shoreline changes.
CHAPTER II
REVIEW OF SURF ZONE MODELS
2 . 1 Review of Existing Longshore Sand Transport Models
Numerous longshore sand transport models exist in the
literature although none give better than an "order of
magnitude" agi^eement with exisiring field and laboratory
data. As the present models are mostly based on wave param
eters as the driving forces, it is important to note that
no sand transport would occur in most of the models if wave
action were not present to both stir up the bottom sediment
and drive the longshore current. Most of the more popular
surf zone littoral sand transport models (2438) are listed
in Tables 2 . 1 and 2.2.
Equation 2.1(a) was historically arrived at by attempts
to relate the volume of sand transported along the coast to
a wave power term referred to as the "longshore energy flux,
P^. Early researchers made empirical formulations of the
type:
= Constant • pj (2.1)
Caldwell (24) first found an expression based on data with
an exponent n  0.8. Other researchers using other data
made separate attempts to refine this equation with various
values of the constant and the exponent, n. Many of these
19
20
Table 2.1. Littoral Sand Transport Models.
Reference
(2.1a) = K^^ (25)
or
(2.1L) = (26)
where = Q^Y(Sg  1) (lp^)
H J. o c , COS a, „
(2.2a) I = K, \ ^ ^ (27)
s 2a w
f s
or 2
YH C , cos CL _
(2.2b) I = ^ V (27)
s 2b w
^ 4/3
cos a, n /o tang sin a,
(2.3) Q, = K3^ — ^ ^^ ^ (28)
d
(2.5) S=i, ti =— ±— .(B (^)^^.S 1)^
^ ^ ^ gdfi ) ^ "hwm ^
V ^ l)d
(2.6) = B*d ^ g= . exp (A* § ) (31)
r^(— ) (1 + — — ) )
z
21
Table 2.1. Definitions.
Kij = constant
= volumetric rate of sand transport (1^/t)
Pj2_ = longshore energy flux (~)
 immersed weight sand transport rate A)
t
3  beach slope angle
= wave breaking angle
w^ = fall velocity (1/t)
= wave breaking height (1)
Y = specific weight of fluid (m/£^t^)
T = wave period (t)
d = grain size (1)
^^b' ^gb ^ "^"^^ speed at breaking (1/t), wave group spe
at breaking
^2 = Chezy friction factor
= ripple parameter = shear of flat sand bed
total shear
A* = empirical coefficient ( = 5 BijkerOl})
= empirical coefficient ( = 0.27 Bijker CSU'^)
C = roughness parameter ( = 0,045 C BiikerOl))
P = fluid density (^/l^)
Pg = sediment density (^/l^)
g = acceleration of gravity Cl/t^)
^hwm ~ ^^^i^um bottom orbital velocity (1/t)
Pq = porosity of sediment in place
S = specific gravity of sediment
22
Table 2.1. Definitions (continued)
K  constant related to ratio of wave breaking
height to depth
E = energy density of waves (5^)
t^
1^  xmmersed wexght transport rate per unit width
(— )
t^'
^s' " efficiency constants
1, t, m = length, time, mass
23
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formulas are sununarxzed iji . TaiJLe. .2 ^ 2 . Later CERC (Coastal
Engineering Research Center) revised the formula with an
exponent n = 1.0 which is now in use as the standard littoral
sand transport equation used by the Corps of Engineers (25).
Limitations in the model exist though due to its neglect
of factors pertaining to the material transported such as
grain size of sediment, density of sediment, porosity of
sediment in place and ■ temperature effects which have been
shown important in suspended load in rivers and can make a
factor of two difference in the transported load where a
wide variation in temperature exists (39). Another short
coming of this formula is that beach slope is neglected.
The constant in equation 2.1(a) is dimensional.
A modification of equation 2.1(a) was first made by
Inman and Bagnold (M0) changing the volumetric rate of
sediment transport to an immersed weight sediment transport
rate as in equation 2.1(b). This allows for a dimensionless
constant (more physically appealing) as well as accounting
for grain density and sediment material porosity effects.
A plot of this equation with considerable laboratory
and field data is shown in Figure 2 of Inman and Frautchy
(25). The dimensionless coefficient K^^ of equation 2.1(b)
was found (26) to be K, , = 0.25. Other researchers have
lb
given various values of K.^^^ equal to 0.2 (32); 0.4 (37);
0.77 (38); and 0.8 (25). Many of these constants are summa
rized in Table 2.1.
25
LonguetHiggins (^.1) has . criticized the above formula
tion as the parameter called "longshore energy flux" is
neither the longshore directed component of energy flux nor
the onshore component of energy flux. As the model does not
differentiate between bed load type of transport and sus
pended load transport, both modes of sediment motion are
included in this model.
Equation 2.2 was postulated in a heuristic model by
Dean (27) on the assumption that sane transport moves as
suspended load. This formula does take into account many of
the missing factors of equations 2.1(a) and 2.1(b), such as
beach slope and temperature (via grain fall velocity), as
well as grain density and porosity of sediment. Additionally
the model can be used for the prediction of sand transport
when a longshore current other than that driven by waves is
present. An additional parameter of importance in this
model is that of a bottom friction factor, C^, which is
dependent on inviscid flow bottom orbital amplitude and
velocity. This factor will be mentioned again later. The
critical shortcoming of this model is its inability to
predict bed load which has been noted by others (42, 38, 43)
to be a major portion of the total sediment transport under
conditions of low wave steepness and large sediment grain
sizes
26
Equation 2.3 was postulated by Iwagaki and Sawaragi
(28) on the basis that the average littoral transport is
proportional to the shear velocity of the mean longshore
current. They use a KalinskeBrown type of sediment trans
port formula. The mean longshore current used in the formu
lation was that of Putnam, Munk , and Traylor (44). Limita
tions in this formula are evident in the lack of grain
density considerations, sedinaent porosity, temperature
effects (due to the use of grain size rather than fall
velocity) and the dimensional form of the proportionality
constant. Density and porosity considerations can be taken
into consideration by a simple transformation used in equa
tion 2.1(b).
Equation 2.U, postulated by Le Mehaute and Brebner (29)
on physical reasoning and empirical information, has the
same limitations as the previous equation along with the
absence of two important parameters, beach slope and grain
size.
Equation 2.5 was postulated by Thornton (30) using an
energistics approach as in Bagnold (45) for calculating the
sand transport per unit width within the surf zone. Thorn
ton's model takes into account most all of the important
sediment transport parameters except for sediment porosity
but does not define the efficiency factors B or S which
s s
makes the equation unusable in a practical sense. Addition
ally, Thornton correlates the work rate of bed load transport
27
to the total available energy flux of the waves rather than
to the energy expended in the longshore direction. As a
result, the energy expended in moving sand back and forth
(in the onshoreoffshore direction) is included within the
available power for longshore sand transport, while, in
reality, no work is being done in the longshore direction by
this energy .dissipation.
Equation 2.6 has been po^stulated by Bijker (31) for bed
load transport in the surf zone based on extensive laboratory
tests and a sand transport model proposed by Frijilink and
modified by Bijker (31). Weaknesses in the model are ex
pressed in the form of empirical coefficients which were
determined on the basis of laboratory tests and give values
ranging over two orders of magnitude. Additionally beach
slope and material porosity are not taken into consideration.
The lack of existence between an expressed form of bed
roughness (in the surf zone) and the Chezy friction factor
adds a qualitative factor to the model which makes it diffi
cult to apply in practical application. Bijker (31) proposes
a suspended load as a function of the existing bed load.
2.2 Review of Longshore Current Velocity Mode ls
As most of the longshore sand transport formulas take
into consideration the longshore current either explicitly
or implicitly, a review of the longshore current models is
pertinent. Most of the longshore current formulas predict
28
only the mean of the longshore current velocity with the
exception of the LonguetHiggins model (46), and the Bowen
model (47).
There are four basic approaches to the development of
predictive equations for currents in the surf zone: conser
vation of mass, conservation of momentum, conservation of
energy, and empirical correlation. Most of the more familiar
equations, are presented in Table 2. .3. An additional long
shore current model by Thornton (30) has not been included
in the table but can be found in Reference (30). Some of
the equations have been expressed in slightly different
forms than originally published in order that they all
incorporate the same parameters; however, these changes deal
only with the geometric relation for the breaking depth and
breaking wave height. The changes do not alter the currents
predicted by the equations Symbols, utilized in the equa
tions are listed in the preface. Much of the following
review of these equations must be credited to Thornton (30).
The equations will be reviewed in order of increasing analyt
ical usefulness with empirical equations having the least
utility due to their lack of physical justification or
rational.
Empirical correlations — Two types of empirical equations
have been developed. The first type employs physical reason
ing to determine the form and grouping of the important
parameters which are then correlated with experimental data.
Brebner and Kamphuis used both the energy and momentum
equations to obtain reasonable groupings by dimensional
analysis of the important parameters. Linear regression was
then used to find the best fit for the longshore velocity to
a large number of data that they had measured in the
laboratory.
The second, type of analysis employs multiple regression
techniques. Sonu et al. (49) used this method to weigh the
various independent variables collected in their field
studies. They found that the most important variable affect
ing the mean longshore current velocity was the angle of
wave incidence, and the second most important, although much
less, was the wind. These results are conflicting with
those of a similar analysis reported by Harrison and Krumbein
(50) who, using data collected at Virginia Beach, Virginia,
found the most important variable to be the wave period
which proved to be insignificant in Sonu's analysis. Sonu
also performed a multiple quasinonlinear regression analysis
which showed the most important variable affecting the mean
longshore current velocity to be the wave height. In a
later study, Harrison (51), using another set of data col
lected at Virginia Beach, found the incident wave angle to
be the most important, followed by the wave period, height,
and beach slope, respectively. Harrison notes that the use
30
of empirical equations is limited in application to "similar"
situations; it is not possible to extrapolate to different
type beaches than considered in the empirical formulation.
Another problem with using empirical formulations is that
they are devoid of physical basis and, as such, can give
spurious correlation and conflicting results .( 52 ) .
Continuity considerationsChiu (53) and Inman and
Bagnold (40) .derived similar ..expressions using the continuity
approach. These formulations are based on the fact that the
incident waves introduce a mass flux of water into the surf
zone which is then manifested as a spatial gradient in the
longshore current. Both developments consider a plane beach
of infinite length, implying that mass is uniformly intro
duced into the surf zone along the beach. The current will
grow (since mass is continually being supplied to the surf
zone), and at intervals, it is necessary that there be
outflow from the surf zone unless the current becomes un
bounded. It is postulated that this outflow occurs in the
form of rip currents which are evenly spaced along the
coast The equations thus contain an unknown parameter — the
spacing of the rip currents. Unfortunately, few measurements
have been made of rip current spacings so that the use of
these equations requires additional experimental data.
It should be noted that, due to the mass flux of waves,
there is always transport of fluid into the surf zone and
31
that,, in all the physical models .whether considering a mass,
energy, or momentum approach, the mass flux must be accounted
for in order to obtain a bounded solution. Thus, the assump
tion that the mass transport is uniformly returned across
the surf zone, is at least implied in all the developments
which do not include concentrated return flow by rip currents.
Chiu and Bruun (53) also considered the case where rip
currents are absent and the return flow is distributed
uniformly over the vertical plane containing the breaker
line. He reasoned that waves breaking at an angle to the
beach contribute mass to the surf zone and locally raise the
mean water level as the breaking wave crest propagates down
the beach. This results in a slope of the water surface
between crests which creates a longshore current. The
longshore current is balanced by bottom shear stress related
to the velocity through the Chezy formula. Galvin and
Eagleson (54) reasoning from the continuity approach,
equated a hypothetical mass flux across the breaker line
proportional to the mass contained in the longshore current.
Using both field and laboratory data, the two mass fluxes
were correlated.
Energy considerationsPutnam at al. (44) also derived
a mean longshore current equation from energy considerations
alone. The derivation equates the changes in energy flux to
the frictional energy losses parallel to the beach. A
32
difficulty, with the resulting equation, is that it involves
two undetermined constants, a friction factor f and the
percentage of the wave energy available to the longshore
current, "s," which makes the equation very difficult to
apply.
Momentum considerations— Putnam, Munk , and Traylor
used both the energy and momentum equations to derive the
first rational equations JLor^^jdescribing longshore currents .
They considered the flux of mass and momentum into a control
volume of differential length bounded by the breaker line
and the shore. The change in momentum flux across the
breakers directed parallel to shore is balanced by the
bottom shear stress. Solitary wave theory was used to
calculate the momentum of the breaking waves . In this
manner, they obtained an expression for the mean longshore
velocity related to the angle of wave incidence , breaking
b
wave height H^^ , bottom slope tanB, wave period T, and fric
tion factor f. Embodied in all of the momentum analyses is
a friction factor that relates the velocity to the bed shear
stress and represents an empirical coefficient. This equa
tion was subsequently revised by Inman and Quinn (52) who
found that a better fit to the data originally collected b/
Putnam et al . , and additional field data collected by the
authors, was obtained if the constant friction coefficient
xn the original equation was changed to be a function of t le
velocity.
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36
Eagleson (55), usi.iig.the same cojitroJ. volume approach,
developed a mathematical model to represent the growth of a
longshore current downstream of a barrier,. In the associated
laboratory experiments , it was found that a large percentage
of the fluid composing plunging breakers (most common labora
tory breaker type) is extracted from the surf zone. This
fluid already has a longshore velocity to which is added the
longshore component of the breaking wave. This argument
provides a mechanism for growth of the longshore current
downstream. The asymptotic solution to the differential
equation showed that the system is stable and that the
growth of the currents was bounded. These results agreed
qualitatively with laboratory results and demonstrated that
unless there are perturbations inducing gradients in the
wave energy in the longshore direction, the current system
tends to be uniform alongshore and stable for stationary
wave conditions .
Bowen (M7) used a conservation of momentum approach to
determine the longshore velocity distribution across the
surf zone for the case of a plane beach with a constant
lateral shear stress. Reasonable results were obtained when
compared to laboratory data. In the case of no lateral
shear stress, his results are comparable to those of Longuet
Higgins except for inclusion of setup.
37
LonguetHiggins ' (46 independently of Bowen, derived
a longshore velocity distribution across the surf zone for
the case of a plane beach based on "radiation stress" theory.
Reasonable results are also found for this theory when
lateral mixing effects are included. As "radiation stress"
theory will be used in the development of a sand transport
equation, this model will be expanded on later.
In the late 1960's Galvin^(56) critically reviewed
the mean longshore current theories and tested those that
predicted longshore current distribution across the surf
zone as well as mean longshore current agreement. Galvin
used both latoratory as well as field data. Galvin concluded
that the most applicable equation appeared to be that of
LonguetHiggins (MS) when proper unknown constant coeffi
cients were used. if
Sonu et al. (49) 'Conducted field experiments and found
poor correlation when compared to six of the above equations.
Their experiments did point out the importance of the near
shore topography on the current system, and how this may
affect the outcome of such results. Shepard and Sayner (57)
pointed out that another reason for lack of agreement between
mean longshore current' theory and field experiments could be
the variation of current across the surf zone; each field
data point is usually based on only a single location in the
surf zone and additionally consists of effects based on a
spectrum of waves rather than one frequency component.
38
Many simplifying assumptions are necessary in developing
the theories. Since exact expressions are not available, it
is necessary, inside the surf zone, to select approximate
expressions for the wave speed, wave shape, water particle
velocity, partitioning of energy in the wave field, a long
shore variation of waves and currents , and velocity and
energy distributions across the surf zone. It is possible
that improved theories for longshore currents will require a
better understanding of the highly nonlinear waves in the
vicinity of the surf zone.
The importance of considering factors other than wave
parameters, such as the wind, was demonstrated by Sonu's
empirical correlations. He found the wind to be the second
most important variable in his set of field data. This
shows the difficulty in comparing tests, particularly field
data where information concerning the effects of bottom
topography and winds often is not included. The extrapola
tion of data from one particular location to another without
accounting for the importance of these effects can lead to
invalid results.
All of the equations involve unknown coefficients to be
determined experimentally. Generally, the friction factor
in the momentum and energy equations is evaluated in the
same manner as in open channel hydraulics. The validity of
utilizing results from steady flow situations without inertia
39
considerations certainly needs to be investigated further
and could hopefully result in a refinement.
Sonu points out that another possible improvement might
derive from consideration of the dynamic processes of energy
dissipation in the surf zone environment. The difficulty is
that the flux of energy used in generating longshore currents
is only a small fraction of the total available energy, and,
as such, represents a second ord.er phenomenon (56). A
difficulty with using the continuity approach is that,
although it allows a description of the mean current, no
procedure has yet been developed, based on continuity consid
erations, which can provide a prediction of the variation of
the current across the surf zone.
Momentum approaches remain the most useful approaches
to date (55) when dealing with longshore currents and will
be used in the development of a sand transport model.
2 . 3 Review of Friction Factor
The longshore current in many of the above formulations
is dependent on bed shear stress which in turn is dependent
on bottom friction factor f ' . The determination of the bed
w
shear stress and hence friction factor for uniform steady
flow has been fairly well established. For oscillatory
flow, and particularly combined waves and currents, the bed
shear stress is not well formulated. This is due primarily
to the lack of good data.
40
Both field and laboratory. data exist for the evalua
tion of the friction factor "f" although field data have
been obtained on an "energy" type approach whereas most
laboratory results have been attained by other means. As
the two approaches provide somewhat different answers
both will be discussed.
2.3.1 Field Data
Field data for friction factor f ' have been obtained
w
by measurement of wave heights at two different locations
and by calculating the energy loss due to bottom energy
dissipation by friction, then equating theoretical energy
dissipation to measured energy dissipation. The friction
factor relates the shear stress per unit area at the bed
T, to the horizontal flow velocity above the bed u, as
h hw
^h = 1 ^^;p%w' ^2.21)
other equations of the general form of equation 2.21 have
appeared with the constant 1/2 replaced by 1 or 1/8.
The rate of wave energy dissipation per unit area, E^,
following Putnam and Johnson (58) is
^f ^h ^hw ^2.22)
Combining equations 2.21 and 2.22 yields:
^f \ P%w' ^2.23)
1+1
All variables in the equations presented thus far indicate
instantaneous values. For the sinusoidal flow at the bed,
predicted by small amplitude theory.
^hw = ^hwm • (2.24)
where u^^^ is the maximum horizontal bottom particle velocity
at the upper limit of the boundary layer (i.e., the maximum
bottom velocity of inviscid f low) . The argument of the sine
term includes the angular frequency, w = 2Tr/T, with T the
wave period and the time, t. Substituting equation 2.214 in
to 2.2 3 and integrating over the wave period gives,
^f = It P %wm^ • (2.25)
Equation 2.25 is the average rate of wave energy dissipation
per unit area of ocean bed due to the frictional shear
stresses produced at the bed.
Bretschneider and Reid (59) give a solution to the wave
energy conservation equation presented in the form of a wave
height reduction factor, K^. This factor accounts for
bottom friction losses in a way similar to linear wave the
ory refraction and shoaling coefficient treatment of wave
energy transformation. Combined they show the change in wave
height due to three effects:
H = ^s ^o • (2.26)
The coefficients of K^, K^, and are the friction, refrac
tion and shoaling coefficients.
42
Br.etschneider (6 0) .using Jield data from wage gages
on two oil platforms in the Gulf of Mexico calculated fric
tion factors averaging 0.10 6 by first calculating the
factor from field measurements and then comparing the
so obtained with the postulated energy dissipation using
the Bretschneider and Reid (59) solution to obtain f.
w
In analyzing bottom losses for waves generated in
shallow water, Bretschneider (5Q) found an average friction
factor of f^ = 0.02. Iwagaki and Kakinuma (51) used signifi
cant wave heights at two stations off the coast of Akita,
with the Bretschneider and Reid (50) method, to estimate
friction factors ranging from 0.066 to 0.180, with an average
of 0.114. Iwagaki and Kakinuma (62) found average friction
factors for 4 other coasts of 0.280, 0.130, 0.110 and 0.100.
The results of various field tests (6063) are given
in Table 2.4, which lists average f values as well as the
' ^ w
range of friction factors measured " in the study. ■ Unfortu
nately, the results of field tests for friction factors are
seen to vary widely which is due to a number of reasons:
(1) additional wind energy effects on wave height in the
measured values of wave height (which could increase or
decrease wave height depending on whether wave energy was
added or extracted from waves by wind between measuring
points); (2) wave frictional damping due to viscid bottom
fluid such as encountered on mud or fine silt bottoms;
1+3
Table 2.i+. Results of Field Measurements of Friction Factor.
Reference
Location
f Average
w
Range of f '
Bret Schneider
(50)
Gulf of Mexico
0.106
0. 0601. 93U
Bretschneider
(60)
Iwagaki
and
Kakinuma
(61D2)
Kishi
(63)
Gulf of Mexico
Akita Coast, Japan
Izumisano Coast
Hiezu Coast (1963)
Nishikinohama Coast
Hiezu Coast (196i4)
Takahama Coast
depths 20MO ft.
slopes  0.006
Niigata , Japan
depth 68 ft.
slope 0.018
0.02
0.116
0.280
0.156
1.100
0.094
0.100
0.035
0.0660,180
0.0540.260
0.5602.320
0.0200. mo
0.0600.160
0.03 0.04
41+
(3) incorrect measurement, .refraction between measuring
points; (4) no dependence on bottom roughness.
The second reason for field data scatter has been
mentioned by Bretschneider and Reid (50) as causing friction
factors an order of magnitude larger than might be experi
enced otherwise in areas with mud bottoms.
The fourth reason mentioned, that bottom roughness was
not taken into account is contrary to what one would expect
as shown in steady flow friction factors from pipe flow
data (64).
2.3.2 Laboratory Data
Jonsson (65) compiled available laboratory data and
conducted additional experiments dealing with turbulent
boundary layers in oscillatory flow. From this study, he
developed a classification of the flow regimes similar to
that for steady flow. The classification is based on a
roughness parameter and characteristic Reynolds number. The
wave boundary layer in nature can always be considered in
the "Hydraulically rough" turbulent regime according to this
classification scheme. '
Unlike the boundary layer in open channel flow, which
essentially extends over the entire depth of flow, the
boundary layer under wave motion constitutes only a very
small fraction of the vertical velocity distribution. This
is because the boundary layer does not have an opportunity
45
to develop under the unsteady velocity field of the wave
action. Above the boundary layer, the free stream is well
described by the potential flow theory (for small waves).
Jonsson using an oscillating flow tunnel to simulate
prototype wave conditions in the laboratory was able to
measure the vertical velocity profile in the boundary layer
and determine the bottom shear stress. He found that for
simple harmonic wave motion .the velordty profile in the
boundary layer could be approximated by a logarithmic dis
tribution and that the instantaneous bottom shear stress was
related to the velocity by the quadratic shear stress
formula
■^h = ^w f ^hl^hl ^2.27)
or
^h = ^w f ^hwm ^°s(a3t8)  coscotS  (2.28)
where ^^vim '^'^^ velocity amplitude of the oscillating flow
just above the boundary layer, 6 is a phase lag, and f is a
w
friction factor associated with the wave motion.
Jonsson also found that the friction factor was practi
cally constant over an oscillation period. The constancy of
the friction factor for particular flow conditions is an im
portant result which allows for a better determination of the
combined shear stress due to waves and the longshore current.
Using the available data from several sources, '.■Jonsson found
■46
that the wave friction factor for laminar wave boundary layer
flow was
f„ = — (2.29)
ew
or, in terms of the friction coefficient as defined in the
energy dissipation approach used in the field studies, is
f; = (2.30)
W /R
ew
where R^^ = wave bottom amplitude Reynold's number
= . (2.31)
For the case of rough turbulent flow Jonsson found that the
friction factor for simple harmonic wave motion could be
represented by an equation of the form
1 . 1 _ „ , . ^h
_ + log — — = 0.08 + log — ~ (2.32)
4 /f 4 Vf~ ^
w w
where r is a measure of roughness, and E,^ is the maximum
horizontal water particle excursion amplitude of the fluid
motion at the bottom as predicted by linear wave theory
^ 2 sinh kh (2.33)
Equation 2.3 2 is based on the roughness parameter r
being equivalent to the Nikuradse roughness parameter (65).
As the roughness is a function of ripple height and spacing
in wave flow, f is also a function of the wave
w
1+7
characteristics. This is because, for the granular beds
consisting of a particular grain size, the ripples adjust
their dimensions according to the wave motion, and it is the
ripple geometry that determines the effective roughness.
Using additional laboratory data, others (6669) have
postulated results similar to Jonsson where the entire range
of wave friction factor f depends on both the wave bottom
w
amplitude Reynold's number R^^ and boundary roughness — wave
amplitude ratio. A summary of some laboratory results for
the rough turbulent range are given in Table 2.5.
Here it is important to again note that f^ and f^ are
not necessarily equivalent as wave friction factor f^ has
been arrived at assuming no phase lag between bottom wave
stress and wave velocity while Jonsson 's data include a
phase lag between maximum bottom wave shear and maximum
bottom wave velocity. . Also, Jonsson 's postulated friction
factor f takes into account '' ■^ bed roughness effects
w
whereas the data used in estimating the friction factor f^
did not account for bed roughness form explicitly.
At present there is only a qualitative understanding
of the ripple geometry as related to sand wave characteris
tics. Generally, the ripples are much more symmetrical in
shape and much longer crested as compared to those found in
alluvial channels. Inman (70) collected a large number of
observations of ripple geometry and wave conditions from
48
Table 2.5. Results of Laboratory Measurements of Friction Factor.
Tp=!t Pnnditions
f
w
Jonsson
Oscillating water channel
0.
020.
30
\03 }
LLlxDUJcii L ijtju.iiu.cix _y xci_y t3X
additional reanalysis of
other laboratory experiments
Tunstall
Wave flume study
0.
090.
50
and
Inman
U, T
2 t; R
Iwagaki
Wave Flume Study
0.
010.
40
and.
laminar boundarv laver
Tsuchiva
(67)
Kamphuis, J. W.
Similar to Jonsson
0.
020.
30
(68)
experiments
Car St ens, et al.
Oscillating flow in channel.
0.
010.
50
(69)
turbulent boundary layer
R = Ripple height.
49
Southern Calif orBia .beacJies  T±i£se ..otiser'vations extended
from a depth of 170 feet to the shore. Since the wave and
sand characteristics vary from deep water to the beach, the
ripple geometry would be expected to vary also. These ob
servations showed that the size of the sand is the most
important factor in determining the geometry of the ripple.
In general, the coarser the sand, the larger the ripples.
Also, there was a general correspondence of decreasing
ripple height with decreasing water depth. The ripples were
smallest in the surf zone where the higher orbital velocities
of the waves tended to plane the ripples off ; the ripples
were almost nonexistent for surf zones with fine sands. The
ripple wave length was related to the wave bottom orbital
excursion. As the bottom orbital velocity of the shoaling
waves increased, the ripple length decreased, increasing the
effective roughness, but, at the same time, the ripple
height is decreased, decreasing the effective roughness.
Carstens et al. (69) in oscillatory flow laboratory
experiments found that the ratio of ripple amplitude to mean
sediment diameter and the ratio of dune amplitude to dune
wave length were found to be unique functions of a single
variable — ratio of water motion amplitude to mean sediment
diameter. If the amplitudetodiameter ratio is less than
775, the ripples are two dimensional with essentially
straight and level crests and troughs. If the amplitudeto
diameter ratio is greater than 1700, the bed is flat
50
regardless of the initial condition. On the intermediate
range from 775 to 1700, the ripples are threedimensional
with ill defined crests and troughs. Ripple amplitude was
found to decrease almost linearly with increasing water
motion amplitude in the range of three dimensional ripples.
CHAPTER III
REVIEW OF EXISTING DATA ON LONGSHORE SAND TRANSPORT
3 . 1 Introductory Remarks
Existing littoral sand transport data fall into two
categories, laboratory data and field data. Data on sand
transport and waves are diffiault to take under laboratory
conditions, not to mention the inherent difficulties in
taking field data. The following review of available data
on sand transport includes manner of testing where available
data exist, and conditions of such testing. The review is
comprehensive to the best of the author's knowledge, although
the data are by no means ideal. In almost all tests of sand
transport many important parameters have not been measured
and very rarely has differentiation been made between bed
load transport and suspended load transport. Much of the
data herein referred to have been used in past studies for
analysis of sand transport models although only order of
magnitude agreement has been found to date. Many of the
laboratory and field tests included here have also been
reviewed by Das (71).
3 . 2 Laboratory Studies
A number of laboratory experiments have been conducted
to determine the mechanisms causing sand transport in the
51
52
surf zone. This discussion is limited to the three
dimensional studies simulating conditions in the prototype.
Krumbein (72) conducted one of the first of these experiments
and concluded that the mean littoral drift was a function of
the deep water wave steepness, H^/L^. Subsequent studies by
Saville (73) showed that a maximum transport occurred for a
wave steepness of 0.025, and, for steepnesses greater or
less than this value, the transport was less. Saville also
found that under low wave steepness much of the sand trans
port occurred on the foreshore of the beach as bed load
while for higher steepness waves much of the sand transport
occurred as suspended load. Both Saville (73) and Savage
(7M) concluded that greater than 9 0% of all longshore sand
transport due to waves occurred within the surf zone (land
ward of breaking waves). This has also been confirmed by at
least one researcher in field studies (75).
Although laboratory results imply that the transport is
a function of wave steepness, the reason for this dependence
has not been established. Galvin (76) conducted a series of
experiments of breaking waves on laboratory beaches . He
developed a classification for determining whether the waves
develop into plunging or spilling breakers as related to
wave steepness and the beach slope. These results, when
compared to the littoral drift studies in the laboratory,
indicate that maximum transport occurs for a plungingtype
53
breaker and that the rate of. transport may be more a func
tion of the manner in which the waves break than the wave
steepness. Battjes (7 7) has defined the parameter which
controls the type of wave breaking as a surf similarity
parameter, c, = "^ ^"^ , where B is beach slope angle, H is
o
wave height, and is deepwater wave length. The smaller
the surf similarity parameter the more the wave breaking
process approaches spilling breaker conditions.
The following Table 3.1 provides a review of laboratory
experiments which will be referred to in later discussion.
Table 3.2 gives the range of wave basins used.
The angles are the nominal settings of the wave
generators to the beach. In the case of "snake" wave
generators, angle is controlled by wave phasing.
A short review of each series of tests follows:
3.2.1 Krumbein Laboratory Study (72)
The laboratory study by Krumbein was conducted in 19142
at the University of California, Berkeley. Longshore trans
port rate, longshore current velocity, and wave and sediment
characteristics were measured. The maximum rate of removal
of sand from the updrift end of the beach was determined
from the rate of feed of material to a hopper. The rate was
adjusted to the capability of the waves to move sand.
Longshore currents were measured by using floats and confetti.
O
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3
55
Table 3.2. Summary of Wave Basin Dimensions.
Source of Data
Krimbein
Saville
Shay and Johnson
Sauvage and Vincent
Fairchild (CERC)
and Savage
Price and Tomlinson
Barcelo '
Bijker
Basin Dimensions
(feet)
58.2 X 38.7 X 2.0
122 X 66 X 2.0
122 X 56 X 2.0
Not Available
150 X 100 X 30
(North Sector)
200 X 150 X 30
(South Sector)
190 X 75
64 X 32
Approximate Beach
Length (feet)
38
60
60 (Counts 2581)
33 (Counts 82123)
Not Available
3098 (Variable)
?
32
?
56
The maximum velocity parafllei to shoi^e near the plunge line
was measured by releasing a soaked string, and recording the
movement of the end of the string over a fixed time interval.
Wave heights were measured with a combination point and hook
gage in which the hookcould be set to the troughs of the
waves and the point to the. crests. The wave height was then
obtained by reading the difference in gage heights on a
single vernier. The mechanical analysis of sand used in the
study gave a median grain diameter of 0.5 mm, although the
sand was not well sorted. Specific gravity of material
(sand) was assumed equal to 2.65. Unfortunately wave breaker
angles were not recorded.
3.2.2 Saville Laboratory Study (73)
Saville conducted a laboratory study at the University
of California, Berkeley. The total transport rate due to
bed load and suspended load was determined by installing a
weighing device at the downcast end of the beach. The bed
load rate was measured by installing hoppers on the beach.
Breaker angles were determined from vertical photographs .
The mechanical analysis of sand showed a median gran size
dgg = 0.30 mm. The sand was relatively uniform in size
with a specific gravity of 2.69.
3.2.3 Shay and Johnson Laboratory Study (78)
A laboratory study was conducted by Shay and Johnson
at the University of California. VJave and sediment
57
characteristics, longshore transport rate, and maximum
longshore current were measured in the tests. Wave height
variability was observed in the experiments therefore many
readings were taken during each run and an arithmetic average
of the readings was used to characterize the wave height,
^ave heights were measured with a point and hook gage as in
Krumbein's tests. The bed load rate was measured by install
ing hoppers on the beach for small generator angles. These
hoppers were removed when wave splitters were used for
angles of 3 0 degrees or greater. Both Bed'load and total
load were measured in these tests similar to those of Saville
Longshore current velocities (maximum and the average)
were measured by releasing fluorescein. The breaker angles
were determined from vertical photographs as in Saville 's
study .
Equilibrium beach profiles were taken after a stable
condition was reached by inserting a piece of sheet metal in
the beach perpendicular to the beach contours and tracing
the sand profile under water with a grease marker, with a
view to study the influence of wave steepness on beach
profile .
The median grain size was d^^ = 0.30 mm and specific
gravity was 2.69.
58
3.2.4 Sauvage and Vincent. T. a, bojatory Study (79)
Two series of laboratory experiments were conducted by
the authors at the hydraulics laboratory at Grenoble, France.
In the first series of tests, the beach was set at an angle
of 15 degrees with the wave makers. In the second series,
the angle was varied from 5 degrees to 70 degrees. In
addition, there was a unique set of experiments in which
three types of sediments with different characteristics were
used. The volume of material, fed from a distributor and
transported along the beach, was collected in a trap at the
downdrift end. The data from the measurements of longshore
transport and the wave characteristics are presented ' only in
graphical form in the paper by Sauvage and Vincent (79).
The actual measured data are not available.
In the studies by Sauvage and Vincent (79), the mean
grain sizes of the three materials used were 0.5 mm, 1.5
mm, and 1.0 mm, and their respective specific gravities were
given as 2.60, 1.40, and 1.1.
3.2.5 Savage and Fairchild Laboratory Study (74, 80)
In a laboratory program on the studies of longshore
transport at the former Beach Erosion Board (BEB), presently
known as Coastal Engineering Research Center (CERC), several
tests were conducted from 19 5 8 to 1965. Ten of these tests
were reported by Savage (74), and all of the tests in edited
form appear in Fairchild (80).
59
The transport studies were made in the north and south
sectors of the Shore Processes Test Basin (SPTB) of CERC. A
detailed description of the SPTB, the sand transporting
system, the sand traps and the sand weighing system is ..in
BEB Technical Memorandum No. 11M, 19 59. The total quantity
of longshore drift caught in the sand traps was recorded for
every 5hour interval during the test.
To establish a r e la tionsiip . between the longshore
energy and the longshore transport rate, the transport rate
between the 20th and 30th hours of test was used, whenever
the test continued over 20 hours.
Due to the large variability in wave heights observed
in the SPTB during the test, it was difficult to characterize
each test with a particular measured incident wave height.
It was decided to characterize the waves by a halfsize
Froude model in another wave flume (75 feet by 1.5 feet by
2.0 foot deep) at CERC. By varying the eccentricity and the
period in the wave flume, the wave heights (before visible
reflection occurred) were measured. The average of these
wave heights was considered as the height of incident wave
at that particular depth of water. The wave height and
eccentricity were derived from halfscale, Froudemodel wave
tank studies.
A family of curves was obtained for wave heights as a
function of wave period with eccentricity as a parameter.
60
These curves were used to diajpacterize the incident waves
in the SPTB (8), and these values (due to the large vari
ability in wave heights in the measurements of wave heights
in SPTB) were used to compute the wave energy in the SPTB.
Fluorescein was used in most tests to measure longshore
currents. Beach samples were also obtained in the tests.'
Water temperature measurements were made during each test
at frequent intervals.
In most of the studies conducted at BEB, no training
walls were placed in the wave basin to conform to the wave
orthogonal for uniform distribution of energy in the beach.
The BEB study also differs from other studies in such aspects
as basin geometry, beach slope, wave height and period
variability, and sand feeder height with respect to still
water level.
Median grain size. was d^^ = 0.22 mm and specific
gravity 2.65 (beach sand). The edited data appearing in
Reference (80) are the results of the averages of various
tests .
3.2.6 Price, Tomlinson, and Willis Laboratory Study (81)
Studies were conducted at Wallingford Research Station
(England) on the effect of groins on a beach stable for a
particular wave condition and beach material. Longshore
transport was also measured on two occasions without groins.
The wave basin is 190 feet by 75 feet equipped with a
snake type wave maker, and tide and tidalcurrent generators.
61
The initial beach slope was. 1 on 17 .at .mean tide level.
The mean grain size and the specific gravity of crushed
coal used in Price, Tomlinson and Willis's studies (81)
were 0.80 mm and 1.35 respectively.
3.2.7 Barcelo Laboratory Study (82)
Studies were conducted at the Portugese National Engi
neering Laboratory on the effect of groins on a beach for
various wave conditions . Longshore transport was measured
in the first hour of the tests by measurement of accretion
volume at a long groin within the basin. Wave heights
measured were an average of random wave heights which cor
responded to a Gaussian distribution. Median grain size
was 1.30 mm and specific gravity equal to 1.6 7 for pumice
stone, the material used in testing.
As data were not listed in (82) except in graphical
form, wave energy values and sand transport values have to
be taken off graphical plots in the report.
3.2.8 Bijker Laboratory Study (83)
The raw data presented in the Bijker (83) report are
sketchy and considerable questionable aspects of the study
are not answered. The longshore current induced in the
model is not driven by waves alone but by a combination of
waves and an imposed gravity slope in the longshore direction.
As a result, the data do not pertain to wave driven sand
62
transport alone but must be the result of additional forcing
functions.
3 . 3 Field Studies
The measurement of littoral drift along a particular
beach and the meaningful correlation with the wave properties
are an extremely difficult field task. Thus, only limited
field data are available.
The earliest published information on the distribution
of sand transport along the beach profile for prototype
conditions was by the Beach Erosion Board (SH). Unfortu
nately, comprehensive wave data wezre not obtained. The
distribution of longshore currents and suspended sand,
obtained from water samples, was measured from piers extend
ing across the surf zone. These measurements showed that
the greatest sand transport occurred at the breaker line,
where the turbulence was a maximum, and decreased shoreward
with another peak in the swash zone — another area of high
turbulence. Seaward of the breakers the sand transport
decreased with increasing depth. There have been few other
field experiments of this type.
Improvements in tracer techniques, particularly using
either radioactive or fluorescent sand tracers, have in
creased the intensity of littoral drift studies in the
field. It is generally assumed that the fluorescent tracer
moves as bed load. A great number of studies have been
63
conducted in recent years on fluorescent tracing and have
been sununarized in a book by Ingle (75). Ingle also con
ducted a number of studies using fluorescent tracers on
several Southern California Beaches. The results of these
studies and some studies conducted by Thorton (30), and also
earlier investigators, are all generally similar to that
given by Zenkovitch (85) and shown in Figure 3.1, describing
the variation of sand transport across the surf zone. The
fluorescent sand grains are found in greatest concentrations
along points of high turbulence such as wave breaking areas .
In a bar trough profile, the sand moves predominantly along
the bar or in the swash zone. There is a minimum of tracer
transport in the trough. During the tests shown in Figure
3.1 the waves were relatively high and spilling breakers
prevailed .
Tracer studies have been conducted using both the
Eulerian and Lagrangian approaches. Most investigators have
used the Lagrangian approach in which the tracer is intro
duced at a particular location, and the concentration dis
tribution of the tracer is determined by obtaining samples
at various sample points. The concentration of tracers is
then determined by counting the tracer grains in the sand
samples. The Eulerian approach is to sample in time along a
particular line across the surf zone, traversing the path of
the tracers. A stable platform or other work facility is
64
Outer Breaker Line
Q 28
4
2
0
75 (m) ICO
150^4
X
i—
a.
UJ
Figure 3.1. Distribution of Longshore Velocity and Sediment Transport
Across the Surf Zone (after Zenkovitch (85"))
8 5
generally required in this method. This was the approach
used by Zenkovitch (85), working from a tramway traversing
the surf zone. Bruun and Battjes (86) and Thornton (30)
worked from a pier.
The inherent difficulty of fluorescent tracer studies
is that quantitative measurements require recovery of most
of the tracer. Unfortunately, the recovery level is gener
ally very low, amounting to only. a few percent. This re
quires that accompanying measurements of the quantity of
sand in suspension, or moving on the bed, also be determined.
However, as a tool, or aid, for solving engineering problems
in which qualitative information can be extremely important,
fluorescent tracer techniques alone can be of great value.
Due to lack of field data such studies are necessary for a
large scale working sand transport correlation until much
more field data are obtained.
Table 3.3 presents a summary of field tests in which
measurements were made of both sand transport and wave
parameters. None of the field tests have differentiated
between bed load and suspended load.
A short summary of each of the various field tests
follows .
3.3.1 Watt Field Study (87)
By measuring wave characteristics, longshore currents,
and amount of material pumped by the bypassing plang on the
66
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67
north side of the north jetty at South Lake Worth Inlet,
Florida, Watts attempted to relate the volume of longshore
transport reaching the pump intake to the wave energy reach
ing adjacent shores. (87).
In the South Lake Worth Inlet region of the Florida
coast, net longshore transport is from north to south. The
material intercepted by the north jetty of the inlet was
bypassed froni the north side to the south shore of the south
jetty to prevent shoaling inside the inlet and to nourish
the south beach. The southerly component of longshore
transport near the inlet was measured from a detention
basin, which was prepared for this study. The sand was
pumped into this basin. During the period sand could not be
pumped into the basin, the pumping rate of the plant was
used to estimate the material pumped.
During the period of study, the pumping plant bypassed
almost all the littoral drift moving alongshore inside the
surf zone. Therefore, it was assumed that the pumped volume
would represent the total southerly longshore transport rate
in the nearshore zone. The material pumped into the deten
tion basin was periodically surveyed for measurement of the
quantity. During the period the detention basin was being
cleared and leveled, the material could not be pumped into
the basin, but the material pumped during these intervals
was estimated from the average pumping rate of 7 6.2 cubic
68
yards per hour, computed from the log of pumping time between
January 1949 and December 1951, furnished by the Palm Beach
County Engineer. Wave heights and periods were measured by
a pressure gage located at Palm Beach Pier, 11 miles north
of the inlet and in about 17 feet of water below mean sea
level. The recording mechanism was programmed to obtain a
12minute record every 4 hours from 6 March to 10 June 19 52.
Wave directions were measured twice daily by the use of
a sighting bar and auxiliary sights attached to an ordinary
engineer transit located on the roof of the Ambassador Hotel
about 3.5 miles north of the bypassing plant.
Significant wave heights and periods were computed from
the wave records from a pressure gage in 17 feet of water.
Frequencies of wave heights and periods were plotted. A
wave direction frequency plot indicated that 7 5 percent of
the direction were from north of east. The predominant
direction of longshore transport along the Florida Coast is
generally from north to south. Therefore, for the data
presented in the analysis, the alongshore component: of wave
energy was computed for each month for southerly wave direc
tions (a<90°) from durations for each a and the corresponding
recorded wave height and period data. The total monthly
southerly longshore transport was evaluated from the pumped
material. The southerly component of wave work was expressed
in footpounds per day per foot of wave crest, and the
69
southerly rate of littoral transport wa.s expressed in
cubic yards per day.
Longshore currents inside the breaker zone were
measured twice daily by using fluorescein at four locations
at distances of 1/2 mile, 2 miles, 5 miles, and 7 miles
north of the bypassing plant at the same time^the wave
directions were measured.
Sediment 'Samples were taken during the study and size
distributions were made by Emery Settling Velocity Tube.
This material was a sand and shell mixture. The data for
sampling stations 1/2 mile north of inlet and 1,000 feet
south of inlet on different dates and at high, mean, and
low tide lines, and at 3foot water depth were presented.
3.3.2 Caldwell Field Study (88)
From measurements of the rate of a longshore sand
movement of a beach fill placed at Surf side, south of
Anaheim Bay, California, and the associated wave charac
teristics, an attempt was made to correlate the two (88).
The beach profile changes shown by seven surveys out to the
20foot contour were used to compute the volume changes of
beach fill along the shore.
Wave heights and periods were obtained from 8minute
wave records taken at 4hour intervals by wave gages in
stalled on the Huntington Beach Pier about 6 miles south of
Anaheim Bay. The depth of water at the gaging station was
70
20 feet below mean lower low. water. . Utilizing meterological
data, wave forecasting techniques were used to "hindcast"
significant wave heights, significant periods, and the
directions of wave approach. Hindcast heights and periods
were used to supplement recorded heights and periods where
gaps occurred in the record. Aerial photographs of the study
area were also taken. Wave direction and period were also
determined from the photographs. Hindcast ing and wave refrac
tion analysis were used in combination to determine the wave
direction associated with each wave observation. The median
grain size in the beach fill area was 0.42 mm.
3.3.3 CERC Field Study (89)
A study was made by the Coastal Engineering Research
Center of the sand deposited behind an offshore breakwater
at Port Huneme, California (89). The sand deposition rate
was estimated by means of detailed hydrographic surveys .
Wave height, period, and direction data were taken by visual
observation at the site and correlated with the sand deposited
in the lee of the breakwater. Unfortunately visual observa
tions were not available throughout the entire length of
the study.
3.3.U Moore and Cole Field Study (90)
A transport rate of 4,6 80 cubic yards per day was
measured near Cape Thompson Alaska, from the growth of a
sand spit during a 3hour period (90). The material was
71
deposited on the spit by waves 5.. feet high with a period
of 5.5 seconds and a 25degree angle to the beach. The'
median grain size was 1.00 mm.
3.3.5 Komar Field Study (38)
Studies were conducted at El Moreno Beach located on
the northwest shore of Gulf of California, in Baja California,
Mexico, and at Silver Strand Beach near San Diego, California
(38). The purpose was to obtain field measurements of the
bed load transport rate over short periods of time and,
simultaneously, to measure the waves and currents to be
able to test relationships between the longshore transport
rate and the wave energy flux.
The measurements of longshore transport rate of sand
were made through use of natural sand colored with a thin
coating of fluorescent dye. The sand advection rate was
determined from the time history of the movement of the
center of gravity of the sand tracer which had been intro
duced onto the beach. The thickness of the sand in motion
was obtained from the depth of burial of the tracer sand in
cores of the beach face. The product of the advection rate
times the cross section of the sand in motion gave the sand
transport rate. The wave direction and wave energy flux
were obtained from simultaneous measurement: of wave charac
teristics by an array of digital wave sensors placed in and
near the surf zone. The wave sensors were of pressure type
72
measuring pressure variation near the bottom. The energy
density obtained from measurements of the pressure trans
ducers was corrected for the damping effect due to over
lying depth of water using the linear pressure response
factor. The rootmean square wave height obtained from the
energy density, and the characteristic wave period obtained
from the frequency spectrum were used to compute the wave
energy flux at the breaker depth. Median grain size was
drn = 06 mm at El Moreno Beach and 0.17 5 at Silver Strand
5 0
Beach, Specific gravity was 2.65.
3.3.6 Thornton Study (30)
In this study bed load measurements were made of sand
transport across the surf zone (30). The traps extended
810 inches off the bottom and were estimated to have an
efficiency of 5090%. The portion of suspended sand
captured by the traps is unknown; therefore, it is questionable
as to whether the entire amount of sand measured in these
experiments is bed load or whether an important percentage
is also suspended load. Wave measurements were made at an
instrumented pier.
CHAPTER IV
SAND TRANSPORT MODEL DEVELOPMENT
4 . 1 Introduction
Past investigations, as discussed in Chapter 2,' have
attempted to relate the mean longshore currents to wave
induced momentum, energy, or. mas.S£liix into the surf zone.
The distribution of longshore currents can be similarly
investigated by considering the changes in the momentum,
energy, or mass flux across the surf zone as has been done
by LonguetHiggins (46), Bowen (47), and Thornton (30). The
present analysis utilizes the momentum principle and the
concept of "radiation stress," the excess momentum flux due
to wave motion (91), to develop the working equations for
longshore current and available power for sand transport.
It is known that, due to the fluctuating water particle
motion of the waves, there is a momentum flux component. If
the waves have a direction component parallel to shore, a
longshore current can be generated due to changes in the
longshore momentum flux component of the shoaling waves
(46). It is also known that there must be a displacement of
the mean water surface elevation to balance the changes in
the onshore momentum flux component of the shoaling waves
(92) .
73
7^
In waves, the momentum flux is the sum of the pressure
and the product of two velocities. It can be shown that the
average momentum flux is nonlinear in wave height. In order
to specify the excess momentum flux of the waves, it there
fore becomes necessary to consider nonlinear, or higher
order effects of the wave motion. Since the waves are
assumed in the present model to be the mechanism responsible
for the generation of the longshore currents , much of the
following discussion will be devoted to describing the
waves .
The treatment of periodic gravity waves is generally
developed by perturbation schemes built on the exact solution
for the linearized equations of motion with specified bound
ary conditions. Difficulty, with this type of analysis, is
encountered when considering higher order theory with non
horizontal bottom boundary. For waves traveling in water of
varying depth, two different approaches can be employed:
the analytical method or the energy flux method. The analyt
ical method involves solving the boundary value problem
accounting for the slope of the bottom to the desired degree
of approximation. This technique includes the slope, explic
itly, in the perturbation expansion. The solution for the
waves on a plane sloping bottom has been worked out to the
second order by several authors (93).
75
The analytical method is generally more consistent
because it accounts for the bottom slope in the bottom
boundary condition to the same order of approximation as is
maintained in the free surface boundary conditions. Small
bottom slope and amplitude are assumed in the perturbation
analysis. Although more attractive from an analytical point
of view, this method has the inherent difficulty of requiring
tedious computations for each.. particular case. Also, the
rigorous boundary value problem approach is not readily
extended across the surf zone.
The energy method consists of solving the wave problem
for a horizontal bottom and then extending these results to
a sloping bottom by means of energy flux considerations.
Hence, for short distances, in the region outside the surf
zone, it is assumed that waves on a sloping bottom can be
considered the same as on a horizontal bottom. Then, adja
cent increments of distance are connected together by means
of the energy flux conservation equation. This results in a
prediction of the wave height at any location outside the
surf zone. Inside the surf zone, it will be assumed that
the wave height is governed by the local water depth.
The energy method allows for the inclusion of energy
dissipative effects, such as bottom friction, and the focus
ing or spreading of energy due to refraction by changes in
bathymetry and currents. These effects are believed to be
76
more important tlian the deformation of the water particle
motion due to the bottom slope as given in the analytical
procedure (93).
The validity of the energy method can be proven by
comparison with the results of the exact analytical solution.
To the first order of approximation in wave energy, for the
case of small bottom slopes, identical results are obtained
for the propagation, of energy. This confirms the use of the
energy flux equation to connect the solution for various
depths for small bottom slopes, at least to the first order
in wave energy.
In the present approach, the energy method describes
the wave field to the first order in energy (second order in
wave amplitude). The solution to the energy equation is
substituted into wave momentum conservation equations to
obtain the working equations needed for predicting longshore
current and hence sand transport.
4 . 2 Review of Mass, Momentum, and Energy Conservation
P'eri'odic Wave Flow '
A starting point for the analysis is a review of the
conservation equations of mass, momentum, and energy fluxes
applicable to unsteady wave motion. The analysis is not
concerned with the internal flow structure of the fluid;
hence, the derivations can be simplified by integrating the
conservation equations over depth. Conservation equations
77
which have already been developed by Phillips (94) will be
used and are presented in a similar format to Phillips.
The conservation equations will be applied to wave
motion, but they are equally applicable to general turbulent
motion. The unsteady velocity field of the wave motion can
be expressed in the same manner as in the treatment of
turbulent motion as the sum of its mean and fluctating
parts 
Resultant Mean Fluctuating
u = U.(x,y,t) + ul (x,y,2,t) ,w(x,y,z,t) (4.1)
where (1,2) refer to the horizontal coordinates (x,y),
respectively, and z is the vertical coordinate. The tensor
notation is used only for horizontal components of water
particle motion. The mean current is assumed uniform over
depth for simplicity. The pressure term can be stated
similarly. These expressions can be substituted into the
mass, momentum, and energy equations, and the mean and
fluctuating contributions identified.
The conservation equations are to be averaged over
depth and time (one can consider averaging over a few wave
periods). For the case of waves superposed on a mean cur
rent, all the wave motion would be identified with the
fluctuating quantity which, when integrated over the total
depth, can contain a mean contribution due to the waves.
The time averaging of the equations for a general development,
78
being over a short interval compared to the total time, does
not preclude long term unsteadiness in the mean motion.
The utility of the conservation equations derived by
Phillips is that the terms involving the mean and fluctuat
ing quantities have been separated. This facilitates the
understanding of the effect of the unsteady wave motion on
the total flow phenomenon. They are a particularly useful
aid in gaining physical . insight _ into the complicated mecha
nisms taking place in the surf zone.
In addition to the conserved quantities of mass, momen
tum, and energy, a fourth conservation equation is available —
conservation of phase (conservation of waves) (95).
The conservation" of phase will be derived for simplic
ity in relation to simple harmonic motion. The expressions
to be derived are also applicable to more general wave
motion which can be formed by the superposition of individ
ual Fourier components. The expression for the water sur
face profile for harmonic motion is given by
1
= 1,2
(4.2)
where j Cx^) is the amplitude, and the quantity Ck^x^  cot)
is called the phase function X. The wavenumber k^ and
frequency to can be defined in terms of the phase function.
such that
■3X
C4 . 3 )
79
An important property of the wavenumber can be seen immedi
ately by vector identity
^xk=VxVX=0 (4.5)
so that the wavenumber is irrotational (see for example
(94)). From equations 4.3 and 4 . M , the kinematical conser
vation equation for the wavenumber can be written
8k. „
1 + 4iiL. = 0 (4.6)
3t 8x
1
For a single wave component, the conservation of phase says
that the rate of increase of the number of waves in a fixed
length is balanced by the net inward flux of waves per unit
time. If (1) can be expressed as a function of k^ and pos
sibly X, from local arguments, (such as assuming a uniform
simple harmonic wave train locally), and further allowing a
mean current U, then,
0) = a(k,x. ) + k.U. (4.7)
a is the local wave frequency and is the apparent frequency
to an observer moving with the current. Substituting into
Equation 4,6, Whitham (95) shows
3k. 3k. 3k.
^ + c . + U. ^^ =  G. (4.8)
3t gj 3Xj : 3Xj 1
where
„ 3U.
G. = + k. ^ (4.9)
1 3x. 3 3x.
80.
and
c . = If (^.10)
c . is the speed at which the energy or values of k are
propagated, commonly called the group velocity. The individ
ual crests propagate with the local phase velocity c = ^.
The notation shall be adopted where, if c or k appear with
out vector notation, it is understood that they represent
the modulus of the vector quantity.
Equation 4.8 says that the rate of change of the wave
number k^, following a point moving with the combined group
and convective velocity, is equal to  G^. Changes in
are due to variations in the mean current and bottom con
figuration. These equations, which state the kinematical
conservation for the wavenumber, hold for any kind of wave
motion (Phillips (94)).
The general conservation of total mass per unit area
can be expressed
^pD._S. .0 i = 1,2 (^.11)
i
D is the total averaged depth of water (which can include a
mean elevation "n, above (or below) the still depth h) so that
D(x,y,t) = (n + h) (4.12)
The overbar is used to signify a time average. The total
mass flux can be partitioned into its mean and fluctuat
ing components
81
M. = M. + M. (H.13)
111
The mass flux per unit width of the mean flow from Phillips
(94) is
M. =
1
_^ pU^ dz = pDU^ (1+.14)
The mass transport of the fluctuating wave motion is
M.
1
n
pu! dz ■ (4.15)
h ^
The equation defining the conservation of horizontal
momentum is derived by integrating the momentum equation,
including shear stresses, over depth and averaging in time.
The balance of total momentum per unit area can be expressed
as per Phillips (94)
—M. + ^ (U.M. + S. . ) = F^. + R. (4.16)
3t 1 9Xj X : X3 px X
Here M denotes the total horizontal momentum per unit area.
X
Hence, the first term on the left represents the rate of
change of the total mean momentum per unit area which in
cludes both the current momentum and wave momentum.
The total mean transport velocity can be expressed in
terms of the total mass flux per unit width as
M. M.
U. = = U. + (4.17)
X pD X pD
The second term on the left of Equation 4.16 expresses the
momentum flux of a steady stream having the same mass flux
82
per unit width and mean .transport veJLojcity as the
actual flow together with an excessive momentum flux term
S^j arising from the superimposed wave motion, where
^ij
^ M.M.
(pu!_u! + p6^j) dz  1/2 PgD^fi^j  ■ (4.18)
h
and is the Kronecker delta. This term is referred to as
a "radiation stress" term by LonguetHiggins and Stewart
(91).
The term F . is given by
Fp. =  pg(n  h) II (^.19)
and represents the net horizontal force per unit area due to
the slope of the free water surface.
R. is the time mean averaged shear stress which must be
X
included in any realistic treatment of the surf zone where
dissipative effects occur. Considering the shear stresses
on a column of water (see Figure 4.1) and integrating over
the depth, R^^ can be expressed
R. =
n n
dz +
dz i,j = 1,2 (4.20)
d Z
where t.. includes the combined stresses of waves and cur
rents. The shear stress on the horizontal plane can be
integrated over depth, such that
Figure 4.1. Shear Stresses Acting on the
Faces of a Water Column
84
85
3t
dz
T . T, .
(4.21)
where the subscripts refer to surface and bottom.
The Leibnitz rule of integration must be employed to
integrate over depth the stresses occurring on the vertical
faces of the water column. Evaluating the shear stress on
the vertical faces and taking the time average of the terms
3t
3 X
il d:
8x,
an 9h
^ji^^ " "^jin ~ ^jih 3x.
h h
Thus, the general shear stress term is given by
(4.22)
R
3x
T . . dz  T . .
31
3n
 T
3h
. . ^ . . ^ + T . T, . (4.23)
3in 3x^ jxh 3x^ ni — hi
h
3 " 3
i,3 = 1,2
The equation for the conservation of total energy can be
partitioned into the energy contributions from the mean and
fluctuating parts with the aid of the conservation of mass
and momentum equations. In the following development, it is
convenient to work with the energy balance' 'for the' fluctu
ating motion alone, which can be stated as per Phillips
(94) .
86
■ Mf . UM? 3U.
In separating the mean and fluctuating contributions , the
mean energy density has been represented as the total energy
density of an "equivalent" uniform flow with the same depth
and mass flux as the actual flow, plus the energy density of
the fluctuating motion, minus an additional term represent
ing the difference between i^ie BTiergy density of the actual
mean motion and that of the equivalent uniform flow. Then,
the first term of Equation 4.24 is the rate of change of the
energy density of the fluctuating motion minus the correc
tion term. The second term represents the convection of the
fluctuating motion F, minus the correction term. The trans
port of energy by the fluctuating motion is given by
F. = P
n
2
ul_ (l/2(u!^ + w'^) + g(z  n) + p) dz (4.25)
h
which represents the rate of work done by fluctuating water
particle motion (for turbulence, this would be the work done
by the Reynolds stresses) throughout the interior region of
the flow, plus the work done by the pressure and gravity
forces. The last term on the left of Equation 4.24 repre
sents the rate of work of the fluctuating motion against the
mean rate of shear. Dissipative effects of turbulence have
87
also been allowed where £ jLs the rate of energy dissipation
per unit area.
The energy budget for the mean motion is given by
1^ (1/2 U.M. + 1/2 pgCn  h^)) +
3S
3I M. (1/2 G.2 . gW) . U 3^ = U.T^. . (4.26)
The terms of this equation represent, respectively, the rate
of change of the kinetic and potential energy of the mean
motion, the transport of the total mean energy, and the rate
of work by mean motion on the fluctuating motion and bottom
shear stresses.
In the development of the conservation equations, no
restrictions were placed on the wave slopes or amplitudes.
Also, no restrictions were placed on the fluctuating motion
so that the equations are equally applicable to wave or
turbulent motion.
4 . 3 Review of Wave Field Equatiohs
The descriptive equations for the wave field are de
rived by solving the linearized boundary value problem over
a horizontal bed. This solution can then be extended to a
higher order by perturbation techniques. The development
presented here will retain terms to the second order in
amplitude (first order in energy and momentum) and neglect
all higher order terms. The wave solution can then be
substituted directly into the conservation equations
88
providing a means .for describing the waveinduced mean
motions. In making this substitution and dropping all terms
of orders higher that the second, only knowledge of the
first order (linear theory) wave water particle velocities
and surface elevation is necessary. This is because, in
expanding and then averaging over the period, the terms
involving higher order quantities in velocity and surface
elevation go to zero. The pressure must be known to the
second order in wave height; however, the average second
order pressure component can be determined from the first
order water particle velocities and surface elevation terms.
Thus, only the linear wave solution small amplitude is
required.
In utilizing linear theory, it is assumed that the
motion is irrotational and that the fluid is incompressible
and inviscid (see for example Kinsman (96)). These have
been shown to be good assumptions (96). The fact that the
linear theory is a good approximation is demonstrated by its
success in describing many observed phenomena (96).
Assuming simple harmonic motion, the surface elevation
is (95)
n = Y (k^x^  CTt) i = 1,2
where the wavenumber components are
k = k cosa
X (4.27)
k = k sina
89
The arbitrary angle of wave incidence a is measured between
a line parallel to the contours and the wave crests. The
velocity potential for the first order solution is given by
(96)
H cosh k(h + z) ^.^ _ (^_28)
2 ck cosh kh x x
The velocity is related to the gradient of the velocity
potential
=  (4.29)
H g i cosh k(h + z) ^^^/^ ^ qn^
U . = TT ^ — r — rr cosCk.x.  at J (.M.ou;
X 2 c k cosh kh x x
The frequency equation relates the frequency to the local
wavenumber and water depth (96)
= gk tanh kh (4.31)
The pressure can be determined as a function of the depth to
the second order by integrating Euler ' s vertical momentum
equation over the depth and retaining terms of second order.
Only knowledge of the first order water particle velocities
and surface elevation is required to determine the mean
second order pressure term. The timemean pressure is given
as
2
— 1 2 tt2 sinh k(z + h) /,.
p =  pgz  gpa H 2 (4.32)
sinh kh
where the first term is the hydrostatic contribution, and
the second term is the second order mean dynamical pressure
component .
90
The group velocity .of ±±ie waves can be determined from
Equation 4 . 9
(c ). = ^ = £ n ^ = c,n (4.33)
g 1 k dk k k 1
where the transmission coefficient is
n = 1 (1 + .^^^ ) (4.34)
" 2 sinh 2kh^
For deep water (infinite depth), n = and, in shallow
water, n approaches 1.
The energy density of the waves is proportional to the
wave height squared and is equally partitioned between the
potential and kinetic energy, such that the total energy
density is
E = ipgH^ (4.35)
Recalling that the group velocity expresses the speed of
energy propagation, the energy flux of the wave motion is
F. = E c . (4.36)
Assuming negligible wave reflection and a mildly sloping
bottom, Equations '.4.3 3 and .4. 3 5 "' can be substituted
directly into Equation :4.35: to obtain the flux of wave
energy.
The mass transport of the waves can be determined by
recalling Equation '4.15. in which the fluctuating velocity
was integrated from the bottom to the surface. Since the
water surface elevation is unknown, the integrand is ex
panded in a Taylor series about the mean water surface level
n
91
M. =
pul^ dz +
h
3ul (n)
p(u!(n) + z rr^ + ...) dz (4.37)
1 o Z
n
where the time mean average of the mass transport below the
mean surface is zero (first integral), and only terms of
second order in amplitude will be retained. Substituting
for the wave profile and water particle velocity of the wave
motion, Equations 4.2, and ■:4.30:, the ftiass transport of
the waves is
TII = 5. _i (4.38 )
M . = 0 nu . —  — T
1 2r\ c k
Simple harmonic motion has been selected to describe the
waves because experience has shown that this solution gives
fairly good results for the deep water case. Inside the
surf zone, however, the approximation is not good, but the
assumption of simple harmonic wave trains will be retained
with the exception of certain modifications on the celerity
and amplitude. The wave field could be specified using the
cnoidal or other higher order wave theories, but the accu
racy gained would not justify the increased complexity of
the resulting equations (93).
To simplify the working equations a determination of
mass transport and energy is first arrived at. The schema
tic of the coordinate system used is shown in Figure 4.2.
32
ELEVATION VIEW
Figure 4.2. Schematic of the Surf Zone
93
Although a longshore current distribution is shown, no prior
knowledge of its actual shape has been assumed at this time.
Due to the absence of any ydependence , the mass con
servation Equation (M.H; reduces to
= 0 (4.39)
3x
Integration gives
M = constant =0 (4.40)
X
The constant of integration is equal to zero since the beach
forms a boundary in the xdirection. This then says using
equation 4.17 that
U =  =  cosa (4.41)
pD pcD
which states that there is a mean reverse current balancing
the mass transport onshore due to the wave motion. This must
be true everywhere, both inside and outside the surf zone,
to ensure that there is no accumulation of mass or growth of
currents in the ydirection in order to maintain steady
state conditions in accordance with the original assumptions.
The determination of the energy distribution is neces
sary as a means of relating the wave heights at various
depths. The energy, in turn, must be related to the local
angle of wave incidence. The angle of wave incidence is
affected by refraction which can occur since the waves are
allowed to approach at an arbitrary angle of incidence to
the bottom contours. The problem is further complicated
9^
.since a shear flow is allowed which can also produce wave
refraction. In the present model this shear flow effect on
wave refraction will be assumed to be a minor effect and
hence neglected. This is comparable to assuming that the
longshore directed current is small in comparison to the
maximum wave orbital velocity.
The general statement of the phase relation gives an
expression r.elating. thje., frequency, mean current motion, and
wave angle. Equation '. M.?' can be expanded to give
(D = cr + Vk sina + Uk cosa (h.^■2)
The wavenumber was shown to be irrotational
3k Sk
^  ^ = 0 (4.43)
3x 3y
Since the wave length and amplitude of the waves are indepen
dent of the ydirection, the gradient of the local wave
number in the ydirection is zero.
3k 3k
Integrating the gradient of k^ in the xdirection gives
k sina = constant = k^sina^ (4.45)
o o
where the subscript "o" refers to conditions in deep water.
This is a statement that, for straight and parallel contours,
the projection of the wavenumber on the beach is a constant.
Assuming zero current in deep water, a general expression
for the celerity of the waves can be derived where, as
before ;
95
0) = + V sina + U casa) (4.46)
and utilizing Equation '!4.45?
c = (—JS. V) sina  U cosa (4.47)
sma
o
For U = V = 0, this expression simplifies to Snell's law for
wave refraction.
The conservation of energy for the fluctuating motion,
Equation ;4.24'', can be expanded for steadystate conditions
to give
. U M ^ U M ^
f(EU + Ec  44 ^4r) + S 1^ + S ^ =  z (4.48)
3x X gx 2pD 2pD xy 3x xx 3x
where the gradients in the ydirection are zero. This
equation states that the change in energy flux, due to
currents and waves plus the work done by the excess momentum
flux on the straining motion, is equal to the energy dissi
pated by turbulence and work done on the bottom. The prod
uct of a stress times rate of strain is a quantity that can
be associated with power per unit volume. The last two
terms on the left of the energy equation can be interpreted
in this context where the excess momentum flux tensor then
represents a stress and the velocityshear a rate of strain.
LonguetHiggins and Stewart (91) named the excess momentum
flux tensor a " radiation stress . "
The excess momentiim flux tensor can be determined by
substituting the wave expressions into Equation '4.18' . In
96
terms of energy, group velocity, and wave speed, an expres
sion applicable to both inside and outside the surf ^one is
given by
c 2c
E^os^a + ^i—^  1)
c 2 c
5. — S. sin2a
2 c
1 S. sin2a
2 c
c c
E sin^a + (2^  1)
c 2 c
(4.1+9)
Effects of turbulence have not been included.
Referring to Equation (4.41) and (4.49), it can be seen
that U and S are of order E. The first and last terms of
Equation (4.48) involving the product of these terms are of
order (E ) and, hence, will be ignored in this analysis.
From the result of the conservation of mass, Equation (4.41),
U = 0, so that the third and fourth terms of the energy
X '
equation are zero. Substituting for S and F , and retain
ing only terms of first order in energy, the energy equation
reduces to
(4.50)
8 N J. n sin2a 3V _
r— (Ecn cosa) + En — ^ ir—   e
dX / dX
where the substitution, c = cn, has been made. Substituting
for the wave celerity, as given by Equation (4.47), yields
(—  V) sina cosa  U cos a
sma
o
3V
+ En cosa sina =  £ (4.51)
d X
97
Again recalling that U is .of. ordBr E, and that all higher
order terms involving the product of U can be neglected in
this analysis, the energy equation can be written
°_ En( . °  V) sina cosa + En cosa sina =  £ (4.52)
3x sm
o
Expanding and cancelling terms, gives
V) I En sina cosa =  £ (4.53)
sma 8x
o
Far outside the surf zone, energy dissipation due to bottom
effects can be ignored, and the energy losses due to turbu
lence can be assumed negligible. Since the term in the
brackets of Equation ..;4.5'3' is nonzero, the result of the
energy equation outside the surf zone, assuming no energy
dissipation, is
En sin2a = constant (1.54)
The relative amplification of the wave energy is then given
by
_ n sin2a
^ = ° . ^ • (4.55)
E n sin2a
o
Since the term involving U in the celerity equation resulted
in the product of higher order terms that are not included.
Equation '4.47' can be written with the additional assump
tion that the velocity V is minor outside the surf zone,
c
c = (~ — ) sina (4.56)
sma
o
or in terms of the wave angle
98
c
(4.57)
sxna =
sma
c
o
o
which is a statement of Snell's Law of Refraction (96).
Equations (4.53; and '4.55:'give a complete description of
the wave amplitude and direction outside the surf zone.
Equation :4.47 can be combined with Equation '■4.55: to
give
This states that the onshore component of energy flux is
constant outside the breaking zone. The changes in energy
density E, or wave height, can be determined from this
equation as a function of the local wavenumber and water
depth. For decreasing depth as the waves approach the
shore, the local wave length and the angle of incidence
decrease. The effect of shoaling is determined by the group
velocity. The group velocity initially increases slightly
so the energy density decreases; the group velocity then
decreases resulting in a continual increase in the energy
density towards shore. A maximum wave height occurs at
breaking. Due to the change in wave angle, which is the
result of refraction, the wave crests become more nearly
parallel to the beach. The energy density is less for waves
approaching at an angle to a constant sloping beach than for
waves whose orthogonals are normal to the beach because wave
refraction results in divergence of wave energy.
E c cosa
g
c
 E — cosa •= const
o 2 o
(4.58)
99
As has been shown, there is an excess of momentum
flux due to the presence of the waves. For conservation
of momentum flux, there must be a force exerted in the
opposite direction such as a hydrostatic pressure force
or bottom shear stress to balance this excess momentum
flux. It has been shown that outside the surf zone the
component of excess momentum flux directed perpendicular
to the contours is balanced by mean water level setdown,
and inside the surf zone that such a balance leads to a
wave set up (97).
These effects have shown (9 7) to have a minor
effect (second order) on water depth outside the surf zone
and for all practical purposes can be neglected in that
region.
The seaward edge of the surf zone is usually de
lineated by the point where the waves first start to break
which is reached when the particle velocity at the wave
crest is equal to the wave celerity. Inside the surf zone,
the waves are unstable, and the fluid motion tends to lose
its ordered character. Waves break in different ways,
depending primarily upon the wave steepness and slope of
the beach. The manner in which they break has a very
definite influence on the hydrodynamics inside the surf
zone which, in turn, affects such quantities as the sedi
ment transport, longshore currents, and wave runup.
100
Based on observations by Galvin (76), breaking waves
are usually classified as spilling, plunging, or surging.
Spilling occurs when the wave crests become unstable, curl
over slightly at the top, creating a foamy advancing face.
Plunging occurs on steeper beaches when the wave becomes
very asymmetric; the crest curls over, falling forward of
the face resulting in the creation of considerable turbu
lence, after which a borelike. wave. front develops. Surging
occurs when the wave crest remains unbroken while the base
of the front face of the wave, with minor turbulence gen
erated, advances up the beach.
There is a continuous gradation in the type of break
ing, and Galvin (76) found it convenient to add a fourth
category of collapsing to describe a type intermediate to
plunging and surging. He performed extensive laboratory
investigations to quantitatively classify breaking waves
according to the wave and beach characteristics . Combining
his results with earlier works, he grouped the breaker type
depending on beach slope tanB, wave period T, and either
deep water or breaker height H. The breaker type can be
established as shown by Battjes (77), by the dimensionless
(H /L )^2 (H^/L^)^^^
variable, — 2 — 2 or , . As either of these
tanp tanp
parameters increases, breaker type changes from surging to
collapsing to plunging to spilling. Spilling breakers are
associated with steep, relatively short period waves and
101
flat beaches; plunging breakers ..are .a^sociated with waves
of intermediate steepness and the steeper beaches; and
surging breakers are associated with waves of small height
and steep beaches.
On natural beaches, breakers classed as spilling are
most commonly observed, followed in decreasing order of
frequency by plunging, collapsing, and surging. In the
laboratory tests, spilling ..bjr>eakers are relatively rare
compared with collapsing and surging breakers because
slopes used in laboratory tests are usually steeper than
slopes commonly found in nature due to the physical limita
tions of space.
The breaking index curve provides a relationship be
tween the breaking depth , the breaking wave height ,
and the wave period T. In shallow water, the relationship
simplifies to H'^^/D^ = < , a constant. LonguetHiggins (M6)
compiled breaking wave data from several sources (9 8106)
including lab and field data. He found a fairly good
correlation for breaking waves as compared to the breaking
wave criteria predicted by the solitary wave theory. The
solitary wave theory predicts a value for k of 0.78, while
other theories predict slightly different values. Theo
retical values range from 0.73, found by Laitone (105), fo
cnoidal wave theory, to a value of 1.0, found by Dean (10 5
using a numerical stream function theory. Experiments on
steeper laboratory beaches show that the value of k can be
102
much larger. All th.e theoretical values have been calcu
lated, assuming a flat bottom, and correspond to beaches
with very gentle slopes. A summary of various investiga
tor's findings zis presented in Table 4.1. The important
point is that the breaker height is governed by the depth of
water .
The energy loss due to the wave breaking process is
dissipated in the generation .ol turbulence and heat, bottom
friction, dissipation, percolation, and viscosity. The
waves in the surf zone constitute a nonconservative system
in which the use of potential flow theory is no longer
valid. In fact, there is no analytical description avail
able for the waves in the surf zone. Gross assumptions are
required. The linear wave theory will be retained as the
input to the conservation equations, but with modification
to the wave amplitude. . The wave height inside the surf zone
as noted is controlled by the depth of water and is of the
same order of magnitude.
Spilling breakers lend themselves to a physical treat
ment since the potential energy and momentum flux of the
«
waves inside the surf zone can be expressed approximately in
analytical form. If the beach slope is very gentle, the
spilling breakers lose energy gradually, and the height of
the breaking wave approximately follows the breaking index
given previously. The height of the wave is then a function
of the depth.
103
H, /2
Table "+.1. Observed and Theoreirical "Values of j = .
Investigator
tanB
K
2
IC
2"
Observed Values
Average
Putnam et al. (44)
0.065
0.098
0.37
0.36
0.139
0.143
0.32
0.37
0.35
0.144
0.32
0.241
0.35
U . Z D VJ
Iverson (98)
0.020
0.41
0.033
0.050
0.38
0.42
0.44
0.100
0.52
Larras (99)
0.010
0.34
0.020
0.091
0.37
0.43
0.39
Ippen and Kulin (100)
0.023
0.60
0.60
Eagleson (101)
0.067
0.56
0 . 56
Galvin and Eagleson ( 54 )
0,104
0.59
0.59
Bowen (47)
0.082
0.450.62
0 . 56
Values Determined
from Solitary Wave Theory
McCowan (102)
0.000
0.39
Davies (103)
0.000
0.41
Long (104)
0.000
0.406
Values Determined by Cnoidal
Wave Theory
Laitone (105)
0.000
0.37
Values Determined by Numerical Wave Theory
Dean (106)
0.000
0.50
104
In the present model, it will be assumed that the waves
act as spilling breakers inside the surf zone and that they
follow the breaking index, k = 0.80. The wave height inside
the surf zone is then given by
H = kD (4.59)
It is further assumed that the kinetic and potential
energy are equally partitioned so that the total wave energy
can be described in .texims of the wave height which is a
function of the depth
E = I pgK^D^ (4.60)
This is a nonconservative statement of the energy distribu
tion within the surf zone.
The waves inside the surf zone are assumed to retain
their simple harmonic character so that the wave profile
and water particle velocity are described by
n = Y cos (k^X^  (jOt)
(4.61)
u = 75 ^ r^ cos (k.x. (jot)
2 c k 1 i
The expression for the horizontal water particle velocity is
based on the Airy wave theory and has been simplified for
shallow water.
In very shallow water, the waves are nondispersive
with the wave speed being only a function of the depth. In
keeping with the linear theory approach to calculation of
wave energy it will be assumed that wave celerity is ade
quately described by linear wave theory
105
c = (gD)^^^ (4.62)
It has been found experimentally in shallow water
breaking, wave celerity can be better approximated by soli
tary wave theory (102);
c = (g(H + D))^''^ = (g(l + <)D)^^^ (4.63)
It is felt though, that since wave height is linear in depth
any way and therefore there is only a difference of a con
stant between the two descriptions of celerity, the linear
wave model should be used for consistency of theory.
M . M Proposed Sand Transport Model
4.4.1, Longshore Current Estimation
The derivation of the longshore current to use in the
sand transport equation parallels an approach similar to
LonguetHiggins (46), and Thornton (30), in that a linear
ized bottom stress in longshore velocity is found from which
longshore current can be postulated. The present approach
is different only in that a wave orbital velocity shear
stress friction factor is used in the bottom shear stress
term, and wave setup is included.
In order to formulate the longshore current problem
some assumptions concerning the flow conditions are restated.
In the present development as in the simplified models of
LonguetHiggins (46) and Bowen C47) it is assumed that
(1) longshore currents are steady, no time dependence,
108
(2) homogeneous and incompressible fluid; (3) hydrostatic
pressure .CLO 7>, l4)rCD'rioliS force is negligible; and (5) currents
are sufficiently small such that wave current interaction is
negligible.
Using the assumption of uniform motion in the y direc
tion and no time dependence. Equation 4.16 can be reduced to
the following forms
0 =  pg (n+h)4^  T, + T
' 8x hx rx
(4.64)
n
h
where
as
T .  T, + T
xydz hy ry
"rx 3x
SS
(4.65)
T
'ry 3x
and
D = n + h (4. 65)
The quantity under the integral sign can be posed
similar ily to a turbulent Reynolds stress as
T dz = y D 1^ (4.67)
xy ex dx
h
Then assuming that the onshore time average bottom
stress is zero, equation 4.6 4 can be reduced to
107
0 = pg(n+.h'>^+ T
rx
(4.68)
8x ex dx hy ry
From equation 4.4 9 with the shallow water assumption it can
be found that
S^^ = I (3  2sin^a) (4.69)
and
c
S = E cosa sina (4.70)
xy c
Also, a relationship for Snell ' s Law of Refraction (i.e.,
Kinsman (96)) is of the form
sina
sma o
= constant = (4.71)
c c
o
which can be included in the above term to obtain
sina
S = F (4.72)
xy X c ,
o
where F is the onshore directed component of wave energy
flux
F = E c cos a (4.73)
X g
and was shown to be constant outside the wave breaking zone.
Therefore, the quantity S is also constant outside the
xy
wave breaking zone and eaual to the deep water value S ,
xy
where
^xy ~ ~2" ^°^^o ^^•'^'^o (4.74)
108
Figure 4.3. Schematic Plan and Section of the Nearshore Region
109
At the shoreline (where D=0) since wave height H is
proportional to depth D the above quantities become
F = 0
X
S =0
xy
and the total external lateral force on the breaking zone
equals
E
Total lateral forc^. on .breaking zone = — sin2a^
(4.75)
Considering now the balance of momentum in the onshore
direction of the water between the breaker line and the
shoreline as expressed by Equation 4.6 8 with the expression
given previously
H = kD = ic(n + h) (4.76)
one can derive the following equation
dx
2(32 sin a, ) ,
K b d
16(n + h)
dx
(n + h)
(4.77)
or
= _ K =  K tang .
dx dx
where
K
2 2
K (3  2 sina, )
b
8 + ic'^O  2 sin'^a, )
b
(4.78)
(4.79 )
If a new horizontal .cGordinate , x, is introduced, with
its origin at the point of maximum setup x = ~^s' "then
X = X + X , and rf + h = (1  K) tan 6 x.
s
Assuming "=0 at the breaker line, one can integrate
equation 4.77 to find n" and the total depth D across the
110
surf zone. In reality, it has .been.shown that there is a
setdown at the breaker zone (i.e., rf = negative value) but
the set down has been shown to be negligible for all prac
tical purposes (97).
Inside the breaker zone F gradually diminishes toward
the shoreline. A consideration of the momentum balance
between two planes x = x^^ and x^ + dx parallel to the shore
line and  separated by a distance dx shows that the net
stress T^^ per unit area exerted by the waves on the water
in the surf zone is given by
^ry =  ^S^y/^^ ^^'SO^
and using equation 4.72 this equation becomes
8F sina
O
3^x
where g— ^ denotes the local rate of energy dissipation in
the onshore direction. This equation notes that the local
stress exerted by the waves is directly proportional to the
local rate of dissipation of onshore directed wave energy.
Outside the breaker zone the mean stress vanishes as S is
xy
constant as shown previously.
In some situations the loss of wave energy can be
attributed to bottom friction (due mainly to the orbital
velocity of the waves). Observations by Munk (108) that in
the surf zone the breaker height is proportional to the mean
depth suggests that under normal circumstances most of the
Ill
loss of wave energy is.due.io wave breaking though not to
bottom friction.
It is important to remember that the large portion of
the wave energy expended in the surf zone is not used in
moving sand in the longshore direction but is dissipated
mostly through the turbulence mechanism. Other investi
gators have correlated the entire energy flux dissipated
throughout the ^surf ^one ■■with the amount of sand transport
moved as suspended and bed load (30), while only a small
fraction of this energy is readily available for actual sand
transport in the longshore direction.
It was shown previously that
H = kD
where k is a constant between 0.7 and 1.0 (see Table 4.1).
Using linear wave theory where kD<<l we have
c = /gD  c
and therefore we have
T tH^ 3/2 ^5/2 ,„
F = ~— c cosa = 5— pg D cosa (4.82)
X 8 g 8 '^'^
and from equation 4.81
3 S
rn xy 5 2 / T^^3/2 dD sina cosa
T =  ^ ^ =  K p ( gD ) 3
ry 3x 16 * dx c
=  <^yD 4^ sina cosa (4.83)
16 ' dx
where ^ = ^ = (1K) tan3 is the modified local bottom slope,
dx
112
Using equations H.82 and the linear shallowwater
theory, we can also express equation 4.83 in terms of the
maximum horizontal bottom orbital velocity given by
^hwm = f '^/^^ = f^"/^^ = f ^ (4.84)
Then we have simply
5 2
=  IT P^hwm ^^^^ sina cosa (4.85)
Seaward of the .breaker line the onshore component of
energy flux is constant (i.e., no dissipation assumed in
the present case) and thus
T =0
ry
The tangential stress excited by the water on the
bottom will be assumed to be given adequately by a relation
of the form
= f f u^ u^ (4.86)
Where u^^ is the instantaneous velocity vector near the
bottom and f ' is a friction factor.
If there were no longshore velocity, and if the ampli
tude of the motion were small and the bottom impermeable,
the horizontal orbital velocity would be expected to be to
andfro in the same straight line, making an angle with the
normal to the shoreline (see Figure 2a in LonguetHiggins
(46) ) .
The frictional stress would then be given by
"^h = f l^hw l^hw (487)
113
which would then vanish ±n ■ the, mean  BJZcoTdlng to linear wave
theory. Here f , has been assumed for f since the motion is
w
entirely due to waves .
If a small component of velocity in the longshore
direction V is added to the orbital velocity, the frictional
stress no longer vanishes in the mean. Assuming a is small,
the component of velocity V is almost perpendicular to the
orbital velociiiy. To first urdeT, the magnitude of the
velocity
l^hl = ^l^hwi' ' \^\^^^^^ (4.88)
is unchanged but the direction of the bottom stress is
changed by a small angle  V/u^^. This leads to an addi
tional stress in the y direction given by
V = J l"hwl l%„l 1:^1 ' ("sa)
Physically, when the orbital velocity is onshore, the
direction of the bottom stress is inclined more tovzard the
positive y direction (when V is positive); when the orbital
velocity is offshore, the bottom stress, now almost in the
opposite directions, is again more toward the positive y
direction. This is shown in Figure 4.4.
Taking the time mean value of the longshore shear
stress the following relationship is found:
114
V (4.90)
hy 2
as is assumed sinusoidal as per linear wave theory
TT ^hwm (4.91)
As the friction coefficient was determined in accordance
with a small V comparatively to the orbital wave velocity
u^^, the bottom shear is dominated by the wave ortibal shear
stress and the proper friction coefficient is assumed to be
f ^ (i.e., the bottom shear stress is governed by the wave
w ^ ^
orbital motion being the dominant shear stress term) .
The friction factor f has been found (65) as a func
w
tion of the bottom wave particle amplitude Reynolds
number and the roughness to wave particle amplitude ratio.
The equation of motion in the longshore direction can
be written as before
8 . ^ 9V.
0 = T , + ^ (y B^)  T, (4.92)
ry 9x ex 3x hy
where in the surf zone T^^ and x^^ are given by equations
4.85 and 4.90, respectively.
Little is known as to the lateral "mixing" coefficient
U which spreads momentum laterally across the surf zone.
A few theories exist, but are based on very limited data
(109). As the spread of lateral momentum across the surf
zone only changes the lateral distribution of the longshore
velocity and its maximum value, and the required final
115
Figure 4.4.
Shear Forces Due to Oscillatory Wave
Velocity and Longshore Current
116
velocity distribution is to ha J.ntagi?ated across the surf
zone, it will first be assumed that y is equal to zero.
This is equivalent to saying that there is no shear coupling
between adjacent water columns taken across the surf zone.
This may be contrary to what various experiments on long
shore currents have shown (109) but will be sufficient for
initial development of the intended longshore current equa
tion. It will lairer be shown that inclusion of an u
value not equal to zero and proper accountability for the
longshore current distribution across the surf zone only
changes the integrated sand transport equation by a constant,
Another way of neglecting the lateral eddy viscosity in
the longshore current equation is to suppose that the ex
change of momentum by turbulence is negligible in comparison
with that due to waves (MS). Then, in general the second
term on the right of equation 14, 92 can be neglected in
comparison with the first.
There remains a balance between the first and third
terms :
T7~ = T (4.93)
hy ry
Substituting from equations 4.8 5 and 4.90, we have in the
breaker zone
f£u^ V = ^pu, ^ (1K) tanS sina cosa (4.94)
TT w 2 hwm 4 nwm
and
117
V = ^J u, (1K) tanB sina 'cosa (4.95)
41 nwm
w
Using the linear theory relationship between and wave
celerity c the equation for longshore current velocity
becomes
V = ~ ^ gD(lK) tang (iiBE) cosa (4.96)
w
where c = /gD as before. By Snell's law the next to last
factor in the above equation is constant. Thus, in this
derivation, the longshore velocity is simply proportional to
local total depth D.
If it is assumed that the shallow water theory is valid
as far out as the breaker line where the depth D is equal to
Dg , the longshore current, in the absence of horizontal
mixing, can be written as
° < °B
0 ^ ^
(4.97)
where
Vg = velocity at breaking line
= 1^ ^ /gU^ (1K) tang sina^ cosa^ (4.98)
w
The velocity distribution represented by equations 4.9 7
and 4.98 is a triangular distribution with a maximum veloc
ity at the breaker line. As noted earlier the turbulent
mixing as represented by a "mixing" eddy cofficient y has
been arbitrarily assumed as zero which provides a
118
discontinuity in the velocity. distri±i.utioji .profile at the
breaker zone.
Both Bowen (47) and LonguetHiggins (1+6) have solved
for the case of lateral shear mixing across the surf zone by
assuming an eddy viscosity based on mixing length theory.
The present extended model exactly parallels the solution
given by LonguetHiggins (46), the only difference being
that wav^ set up is accounted for. In LonguetHiggins
solution (46) and the present solution y is assumed to be
given by a form of mixing length times mixing velocity
= Npx /gD (4.99)
where N is a dimensionless constant; x is a mixing length,
increasing linearily from the shoreline boundary similarly
to the Prandtl hypothesis; and, the mixing velocity is
proportional to the horizontal water particle velocity which
is proportional to wave celerity = /gjD. The solution of
equation 4.9 2 under the consideration of lateral shear
mixing is given in (46) and the solutions are shown here
with the modification that wave setup is included. The
equations for longshore current distribution with wave set
up are
V„ (B.X ^1 + AX) 0< X<1
V = { ^ ^ p (4.100)
Vg (B2X^2) 1< X < <»
and by matching solutions at the breaker line
119
x=l
V
x=l
8x
x=l
3 x = l
3x
(4.101)
the following values are found
1/2
. + L>
4 '16 p
= . 3 , (4 , 1,
„ _ 3 , 9 , 1.1/2
B
B,
A
. A
and where
P2I
P1P2
Pl1
P1P2
2_
2Sp
2ttN(1K) tanB
KT
(4.102)
(4.103)
(4.104)
(4.105 )
(4,106 )
(4.107)
w
X = x/x, a dimensionles s distance
x^ = distance to the breaker line from the shoreline
As can be seen in the above equations at the value of p = 2/5
a singularity exists in the above solutions. The solution at
p = 2/5 is of a separate form (46), and is given by
0<X<1
(4.108)
l<X<oo
V =
Vg (10/49X  5/7X In X)
Vg (10/49X  5/2)
As before, Vg equals the longshore current velocity at the
breaker zone in the absence of lateral shear mixing and is
defined in equation 4.98.
120
Experimental evidence in laboratory longshore current
experiments (46) suggests a coefficient of p between 0.1 and
0.4 when comparing laboratory longshore current measurements
to theoretical values.
4.4.2 Integrated Sand Transport Estimation
The proposed sand transport model is an extension of an
existing model of sand transport based on an energyrtype
approach due to Bagnold (45) and postulated for unidirec
tional current flow. The present model extends Bagnold 's
basic concept to the realm of sand transport along beaches
parallel to the shoreline due to wave action. The concept
can be extended to the case of sand transport along beaches
due to other driving forces such as wind shear on the surf
zone or shear due to tidal induced currents (i.e., hydraulic
tidal currents in the vicinity of inlets). In the presently
developed model though ' only the sand transport due to wave
motion will be considered.
The presently developed model also differentiates
between bed load and suspended load sand transport and each
component can be obtained independently, a distinct advan
tage not realized in many sand transport models (i.e.,
Einstein (110)) where the suspended load component is
dependent upon the calculated bed load component.
Now, defining the dry mass of a unit area of material
to be m and the solid density of the material to be p , the
121
immersed weight of the tmit . ar ea of ied material is
m (— ^ — )g. This immersed weight of sediment can be divided
PgP
into two parts m^ and where the symbol m' = m(^ ).
The b.ed load m^ is that part of the load which is sup
ported wholly by a solidtransmitted stress m^g and the
suspended load m' is that part which is supported by a
fluidtransmitted stress m'g. The total transport rate for
the immersed mass of material per unit width of bed is
H = "bs ' ^ss h ^^^^^^
where
iTj^ = average velocity of the sand particles in the bed
load
and
U = average velocity of the sand particles in the
suspended mode of motion
i^ = bed load immersed weight transport rate per unit
width bed
i = suspended load immersed weight transport rate per
^ unit width bed
The dynamic transport rates i^ and i^ have the dimensions
and quantity of work rates, being the products of weight
force per unit bed area times velocity. As these quantities
stand, they are not in fact work rates though for the stress
is not in the same direction as the velocity of its action.
The dynamic transport rates become actual work rates
when multiplied by conversion factors and A^ , each
defined as a ratio
122
tractive stress needed ±q Tnainj^iTg transport of the load
normal stress due to immersed weight of the load
(4.110)
For bed load the factor A^^ is a friction coefficient
tan(}>, where tan(j) expresses the ratio between normal stresses
on the bed and lateral stresses opposing motion parallel to
the bed (which is some portion of the normal stress as in
solid physics). The forces acting on a unit area of bed are
as shown in Figure 4.5.
The analogy in solid physics can be expressed with a
solid body on a flat surface. The coefficient of solid
friction at rest is most easily measured by resting one body
upon the other and increasing the angle of inclination of
the shear plane until shearing begins by gravity. Then
since gravity force parallel to slope = mg sine}) , and
gravity force perpendicular to slope = mg cos^ , the coeffi
s
, parallel force ^ a j. j. ■ c
cient — —f — r. =. ^ = tan({) = static angle of repose
perpendicular force s ^ ^
of sand.
In solids the dynamic coefficient of friction is lower
than the static coefficient of friction; therefore, the
expressed angle cp would not be expected to be equivalent to
the angle of repose except for the limiting static case
which provides an angle of repose = 3 3° for which tan(j) = 0.63.
s
Bagnold (45) found that the dynamic friction coeffi
cient or stress ratio across shear planes of an array of
solid grains is of the same order as the static coefficient
123
Lift = L
Prog
Weight=G,
(G3L5) tan<^
GsLs =Dispersive Stress due to
Normal Grain Momentum
G.U
Friction Force Resisting Movement
of Sand Grains
/
/
/
/
= (GsL3)tan<^ CDrcg
(for Sediment Movement)
Figure 4.5. Relation of Normal Force to Moving Force
124
when the grains are closely packed and also when they are
considerably dispersed. The actual value of tan(}) was found
by Bagnold (45) to be a function of a parameter for
dispersed solids in a fluid and is analogous to a fluid
Reynolds number where
^s = f
S U ^
and X is a linaar . spacial. concentration defined by the ratio
X  mean diameter d of solids
mean free distance between solids
The bed' load work rate then becomes
bed load work rate = m'g U, tan<J) = i, tan4> (4.111)
b° bs b
since the work rate must be the resistance force (in the
direction of the moving grains) times the velocity of the
moving grains .
The suspended load work rate can be inferred more
simply. The suspended solids are falling relative to the
fluid at their mean fall velocitv w , but the center of
s '
gravity of the suspension as a whole when time averaged does
not fall relative to the bed. Thus the fluid must be lift
ing the solids at the velocity w . The rate of lifting work
done by the shear turbulence of the fluid must be
^s
suspended load work rate = m'gw = i, — — (4.112)
S S D ^7
ss
which points out that the factor A is equivalent to
125
w
A =
= U
ss •
When any kind of continuing work is being steadily done
the principle of energy conservation can be expressed in
terms of the time rates of energy input to, and output from,
a specified system by the equation
rate of doing work = available power  utilized power
or in an equivalent alternative form
rate of doing work = available power x efficiency (4.113)
d Work
dt
= P
For bed load transport this equation becomes
±^ tan<) = % ' '^■^ (U.im)
or
e,
= :rf—r (4.115)
b tantp b
with e^^ an efficiency factor for bedload (as defined in
Bagnold (45)) less than unity. As the bed load force used
to determine bejilload work rate is in the direction of trans
port, the available power P^ for bed load transport in the
longshore direction refers to the power expended on the
bottom in the longshore direction.
The power used to support the suspended load is e P
s s
therefore, the equation for the suspended load transport
component becomes
126
w
i — = e P ■ (4.116)
s u s s
ss
or _
U
ss T3
1 = e P
3 8 W S
S
The proper quantity of available power to use in this case
P is the power available for suspension of the sand, not
5
the longshore power as in the bed load. More will be men
tioned concerning this power quantity later.
Adding components of transport, the equation for bed
load and suspended load transport combined becomes
e. e U
i = i, + i = (3^^)P, + ( ^ ^^ )P (4.117)
b s tanc}) b w^ s
It will be assumed that the mean velocity of the suspended
solids U equals the mean velocity of the fluid which for a
steady longshore current becomes V.
It remains to discuss the available power in terms of
wave motion and the efficiency factors for the various
transport modes.
Available power for transporting sand in the longshore
direction for bedlload is that power dissipated by the bottom
stress in the longshore direction. In the analogous case of
riverine sand transport the power dissipated would be equal
to
Available stream power = x, U
127
where x^^ is the bottom stress per unit area of bed and U is
the average velocity of the stream.
For the case of wave motion in the surf zone the power
P, becomes
D
Available longshore power
for bedload transport = t^^ • v = (x) (4.118)
where x, = the bottom stress exerted by the combined wave
and current action ±n the longshore direction
V = the longshore velocity
In the case of no lateral mixing the longshore wave induced
stress would be from equation 4.93.
3 S
T  _ S 2 ^ _,>,3/2 T,v, „ sina cosa , n \
hy " 3x 16" P^g^* (1K)tan3 (4.119)
and as shown previously in equation 4.9 6
V = ~ J /gD^ (1K) tanS sina^ cosa^ ~ (4.120)
w B
Using equations 4.118, 4.119, and 4.12 0 and invoking the
shallow water assumption and Snell's Law, the total long
shore power available for bed load transport becomes
P^(x) = i(i)p . (gD)^^^ • (1K)^tan^g( ^ ^ )^cQsa^ cosa
w b
(4.121)
For the case of suspended load sand transport the
analogy between the available power in riverine system and
the surf zone no: longer holds . For the riverine system the
128
available power for suspension of sand is the same as the
available power for bed load motion. For the surf zone
though, complex processes act to entrain sediment into the
flow system such as breaking waves plunging on the beach and
creation of turbulence by surging wave action ( similar to
"bores" in a river). In this case the available power for
suspension is the total power expended by the waves in the
surf zone. .Th,usy. at a,, spexiific .location in the surf zone
the available power for suspension of sand is
P = I (Ec cosa) (4.122)
s dx g
where Ec represents the energy flux per foot of wave crest
and the cosa factor resolves the power per foot of beach
rather than per foot of wave crest. The quantity P then
represents the onshore gradient of the onshore energy flux.
The total sand transport due to bed load and suspended
load is, from equation M.115 with U = V
s s
e, e V
i = i, + i = (t^^) P, + (— ) P^ (4.123)
I h s tan(J) b w s
s
where now
^hy
(x)
25tt
3
IC
16
8f
w
V(x)
s inct
(gD)^'^^ (1K)^ tan^S (— — )'^ cosa^ cosa
b
(4.124)
and
P (x) = I (Ec cosa) = T<^T ^ D^^^ (1K) tang cosa(4.125)
S dX g 16
129
Upon integration of these quantities across the surf zone
the total transport becomes (assuming the efficiency factors
are constant across the surf zone)
^t) s e
I. = / (i^ + ie) = ^irr) P>,(x)dx+^ VP^(x)dx
X o D s tanqj o d , w o s
/•■I o sK sin2a, e^c, cosa,
= (K <(1K)tang ^( b b ^ _s_b ^).p^ (1+.126)
w f tan*' 2 w I
w ^ s
where
yH, 2 sin 2a,
and K = constant = 1.40
w
which can be rewritten as
where
/n o s, sin2a, e„ c, cosa,
= (<(1K)tane) . (_b b ^ _s_^ b) (4.127b)
^ r tan4) • 2 w
w ■ s
At this point it is interesting to compare equation
4.126 with the "modified" presently used sand transport
relationship of Reference (25) discussed earlier,
where
yH, 2 sin 2a,
■n b b
^5, 8 ^b 2
Equations 2.1 and 4.12 7 can be equated by the relation
ship
130
tanip 2 w
s
CK, — ^ tan3)
w X
W
K X
(1+.128)
Using "rough" values of efficiency factors with e^^ = 0.13
from an average of Bagnold's (45) range of values for bed
load efficiency in rivers and e = 0.002 "postulated" in a
heuristic model for suspended sediment transport by waves
(Dean (26)), and with the following assumed values for
typical wave and sediment parameters
tan() =
0.
63
=
0.
20 ft. /sec. (d=0
^b =
2
feet
^b ~
8°
f 
w
0.
02
tan3 =
0 .
01
a value of
^Ib
can be found
^Ib"
0.
50
As noted
in earlier, the
sand transport
computations is
'ib ~ ^'^ although many inves
tigators (111) believe that the present value used is too
high and should be on the order of 0.M0.6 based on
reanalysis of past data.
It therefore appears that the presently postulated sand
transport formula appears to provide a reasonable approxi
mation for the prediction of sand transport on beaches.
This will be checked out further later.
131
4. '1. 3 Sand Transport Distribution Across Surf Zone
Now, using the previously developed (46) distribution
of longshore current velocity as given in equations 4.100
and 4.10 8, and the presently developed sand transport
relationship given in equations 4.12 3, 4.124, and 4.12 5, the
distribution of sand transport in the lateral direction
(across the surf zone) can be calculated.
. .Again the equation for' sand transport can be written
i . i + i = ^ . (T V) + !il 8(EPg cosa)
X, b s tan<p hy w^ 3x
where from equation 4.9 0 ~^y t)^ written
T,„ = f , (J /pT /X) V (4.129)
hy w 2 TT ° D
where X is — , a dimensionless distance from the shoreline
^b
and
°b ^b
^b TlK) tang ^ <(1K) tanB (4.130)
This equation can be written in terms of the dimensionless
variable X as
, _ ^b ^ ^^w'^b „l/2„2
P — ^ '
I tan^ ^ 2¥
+ — (y c, D, (1K) tanBcosa) X^^'^V (4.131)
w 15 b b
s
Now the above mentioned longshore current model (which
incorporates lateral mixing) as discussed in equations 4.100
and 4.108 can be used in the above expression. It should be
132
noted that outside the surf zone there can still be bottom
friction in the case of lateral shear mixing, as the bottom
friction balances the gradient in lateral shear force be
tween adjacent water columns (since the gradient of the
"radiation" stress is zero). Hence, bed load can exist
outside the surf zone. As the available power to suspend
the bottom sediment is considered to be due to the wave
breaking process though, there Is no suspended sediment
transport outside the breaking zone.
Using equations 4.98, 4.12 3, 4.124, and 4.12 5, the
equation for sand transport as a function of the distance
across the surf zone can be written as (after considerable
manipulation and use of the shallow water assumption, along
V
with a dimensionless longshore velocity defined as V = tj— )
B
e, sin2a, i m
^% Han(i)2
e^c^cosa ^3/2^ ^25tt^ ^ K:(lK)tanP ^ ^ (4.132)
w 16 f x,
s w b
which can be written in dimensionless form as
777TT£K7tanF7 ^16 ' ^ tan(j)2
f ^
w
e c, cosa, a. ^ / o
^ V X^'^^} (4.133)
w
s
The distribution of sand transport across the surf zone
can now be found using assumed values of the efficiency
133
factors and other appropriate wave and sediment parameters.
As an example the following values are considered:
e, = 0.13
D
tancj) = 0.63
W :
S
■ 0.
20 ft. /sec.
e :
s
: 0.
002
: 2
ft.
'
: 8°
: 0.
2
A dimensionless plot of the sediment transport lateral
distribution for the above assumed values is shown in Figure
1+.6. As referred to previously, there is no allowance for
suspended sediment transport outside the surf zone in the
present model since the energy dissipation (available power
for sand suspension) is zero outside the region of breaking.
This leads to discontinuity in the sand transport distribu
tion at the surf zone. The question as to whether such a
distribution occurs in reality cannot be answered as no data
existjwith which to compare such distributions although at
least one researcher, Ingle (75), found that little sand
transport occurred outside the surf zone during flourescent
sand tracing studies. Other researchers (73, 7 4) make
reference to the absence of sand transport seaward of the
wave breaking area based on laboratory results although no
quantitative data aire given . From color aerial photography
134
of the surf zone it also appears that the breaking wave zone
provides a clear demarcation between the "colored" water
suspended load area inside the wave breaking zone and the
"clear" water area outside the wave breaking zone which is
at least qualitatively in agreement with the present model
(i.e. lateral mixing of sediment seaward of surf zone is
negligable).
Using the postulated sand transport lateral distribu
tion, the sand transport can be integrated across the surf
zone (assuming a mixing coefficient value) and a "refined"
value of coefficient can be found. Again using equation
4.131 and integrating across the surf zone a dimensionless
equation is found;
, 25tt ,
K(lK)tan6^ 16 ^
•)
e, sin2a, ,
tancp • 2
w
(4.134)
e c,cosa, ,
w
which can be rewritten as
I
^ ic(lK)tang ^
f ^
w
(^)
^16 ^
e, sin2a, .,. ^ e c, cosa, j'
(h ^)(i" + t") + ( s ^ ^ ) i"
^tant})2 ^^^i^^2^ ^ w ^ ^3
(4.135)
where
135
136
I* = /^V^X^'^^dX (4.136)
o
I* = TV^X^/^^X . (4.137)
^ 1
I* = /^VX^'^^dX (4.138)
o
using the solutions for V presented in equations 4.100,
these integrals can be integrated to provide the following
values
for "p" arbitrary (but i 2/5)
* B^^ 2AB^ 2A^
^1 = (ip^+3/2) ^ (p^+5/2) ^ T" (4.139)
ft "^2
^2 ^ 2p2+2 (4.140)
for "p" = 2/5
ij = 0.0594
I2 = 0.0139 (4.142)
I* = 0.1166
The values of these integrals have been tabulated and
are presented in Table 4.2.
137
Table 4. 2. Integrals for Sediment Transport.
A
A
p
V^2
T
^3
• UU±
OC A A
A A O C
A C A C
. 2635
A T A C
. 2706
• Quo
A 1 1 A a
. 2428
A A C O
.0058
A P 1 O
. 2486
. 2603
A 1 O T
. 2137
A A A A
. 0092
A O O A
. 2229
A 1 1 T A
. 2419
.1543
AT 1 ) 1 1
. 0144
. 1587
A A A C
. 2005
10
T A A O
. 1228
. 0158
n o A c
. 1386
.1758
. 15
. 1037
.0160
.1197
. 1596
. 20
A A A A
. 0902
A T C A
. 0158
1 A A A
. 1050
.1474
. 25
A T n n
. 0799
. 0154
. 0953
.1377
. 30
. 0718
. 0149
A A
. 0867
.1296
.35
.0651
. 0144
. 0795
.1225
. 40
.0596
. 0139
. 0735
.1167
. 45
.0547
. 0134
. 0681
.1113
. 50
.0505
. 0129
. 0635
.1055
. 55
A 1 1 T A
. 0470
. 0124
A C A 1 1
, 0594
T AAA
. 1022
. 50
. 0409
. 0115
. 0524
. 0948
. 70
. 0384
. 0110
. 0494
A A T C
. 0915
. 75
. 0361
AT A
. 0105
A 1 r T
. 0457
A A O C
. 0885
. oO
A A 1 1 A
. 0340
A T A A
. 0103
. 0443
A O C "7
. 085 /
. 85
. 0322
. 0099
. 0421
. 0831
.90
.0304
.0095
.0399
.0806
.95
.0289
.0092
.0381
.0783
1.00
.0274
.0089
.0363
.0752
1.20
.0227
.0078
.0305
.0687
1.50
.0175
.0064
.0240
.0601
3.00
.0071
.0031
.102
.0373
6.00
.0024
.0012
.0035
.0214
10.00
.0010
.0005
.0015
.0137
138
Estimation of .£f f i ciBiicy Factors and Dynamic Friction
Angle
It remains to discuss appropriate values of the effi
ciency factors e^ and e^ for the case of sand transport and
the dynamic coefficient of friction for bed load movement.
The suspended sediment efficiency factor e^ has been
defined previously as
work rate necessary to maintain a suspended load
s available wave power per foot of beach
(14.143)
S
3x —5 • c • cosa)
0 g
= Ss1
where m' of equation 4.109 = C m^ (— f — )
s ^ f Sg
and C = time and spacially averaged (over depth) dry weight
concentration of sediment
mjT = mass of fluid in a unit area of surf zone at a given
depth = pD
Thus e^ can be reduced to the following equation
e = ^ s (4.iLm)
S o 2
c cosa(lK)tanS
16 g
To determine a value of the factor e^ the problem becomes
one of postulating a method for determining the time and
spacially averaged concentration over depth using wave and
sediment parameters. Heuristic models will first be pre
sented to define the important parameters and then a
139
semiempirical approach based on existing observations of
suspended load in waves will be used to formulate a model
for finding the concentration.
Assume first that a breaking wave crest suspends mater
ial to a maximum height in the water column which will be
considered proportional to the breaking wave height = 6H as
shown in Figure Now, if the time required for the
grain to fall back to the boirtroTii t. = — is less than the
b w
s
wave period T, then the depth averaged concentration as a
function of time, as well as the depth and time averaged
concentration will be shown by Figure 4. 7b. If the time
required for the grain to fall back to the bottom t^ is
b
greater than the wave period (where the dominant crest force
is responsible for suspending the sediment) the concentra
tion will be shown by Figure ^.Ic, with higher depthtime
averaged concentrations than the previous case. It is
therefore apparent that the depth time averaged concentra
tion should be a function of the parameter This
^s
heuristic argument should be kept in mind as a more for
malized approach is now used in an attempt to arrive at an
important parameter for use in prediction of suspended sand
concentration in the surf zone.
It is important at this point to look at the findings
of some investigations of suspended sediment in the surf
140
Figure 4.7. Sediments Suspension Due to (b) Long Periods, High Fall
Velocities, (c) Short Periods, Low Fall Velocities
141
zone. Watts (112) conducted field studies using an elabo
rate continuous suspended sediment sampler from a pier. His
results showed that the amount of sand in suspension was
related to the wave height, or energy, of the waves for a
particular test. In these experiments, and some by Fukushima
and Mizoguchi (113) using suspended samplers made of bamboo
poles, the vertical distribution of suspended sediments was
also measured. These data showed that the amount of sus
pended sediment can be fairly well described over the ver
tical by plotting concentration versus elevation on a semi
logrithmic plot for the case of spilling breakers. Fair
child (IIM) using field equipment similar to that of Watts
and measuring suspended sediment from piers found results
similar to Watts. In most all cases where plunging breakers
were not encountered, the concentration of suspended sedi
ment was found to be exponentially distributed over the
vertical water column. Fairchild (114) also found this
exponential type distribution in similar studies conducted
in a wave tank. Homma and Horikawa (115), Shinohara et al .
(116), Hattori (117), and Kennedy and Locher (118) have all
found this exponential type distribution under shoaling
waves which is suggestive of a constant "sediment exchange
coefficient" in the vertical. This appears to be particu
larly the case for spilling breaking waves (116). Reviewing
the equilibrium exchange equation for sediment in the
142
vertical water., column as given Ro.ose (64) where (using
time averaged values as denoted by a single bar over the
quantity, and primes as fluctuating quantities)
w C(z) = £ = V C (4.145)
s s dz p
where £ = vertical sediment exchange coefficient analgous
s
to the momentum exchange coefficient or eddy viscosity, and
is the sediment particle random velocity. If the exchange
coefficient £^ is independent of the z direction, then upon
integration an equation can be found of the form
C(z) J ^s^^^a^
exp < 
Ciz)
a.
£
S
(4.146)
where C (z ) is a concentration at the level z above
a a
the bottom. Since z is an arbitrary distance it. can be
a
assumed that z^ corresponds to the top of the bed load zone
(Z = 0) and the concentration C (z) can be depth integrated
from z = 0 to z = D giving
_ w D £
C = C (0) (1  e — ~) —fr (4.147)
£ w D
s s
A method for attempting to rationalize out a parameter
of importance in suspending sediment ( "entrainment" ) to a
level near the bed (in order to empirically obtain a value
of ^(0) and hence C) is to consider the forces on a grain of
sand in the near vicinity of the bed about to be lifted out
into the flow (see Figure 4.8).
143
The lift force on a given grain "L" can be expressed as
S L Z 4
and the gravity force weight G action on the particle is
s
expressed as
3 3
= (p^ p) g = pCSg  1) g (4.149)
Now the ratio of lift to weight is
2
L 3C, u,
s L hw
G_ " 4CSg  Dgd
(4.150)
As u^^ is oscillatory, the time average value of the lift to
gravity is
L 3C, u, ^
G  8(Sg  Dgd C4.151)
s
In linear wave theory the maximum velocity (just outside the
wave boundary layer) is
u, (D) = I ( ' I ,r. ^ (4.152)
hwm T smh kD
and the lift to gravity ratio can then be expressed as
r 3c,
s _ L
H IT
T sinh kD
^ =^ (4.153)
G 8KSg  1) gd
o
As grain size pertains to spherical particles in the
above formulation a more general formulation applying to non
spherical particles can be made by expressing the grain size
144
Figure 4.8. Vertical Forces on a Grain Particle
About to be Lifted from the Bed
145
in terms of its fall velocity w by a balance of drag force
to submerged weight of the grain. This balance gives the
following equation for a spherical particle falling at its
terminal velocity
w = (i  1^ gd)l/2
"s ^3 ^
(4.154)
where is a drag coefficient. Now equation 4.15 3 can be
rewritten as
2C
D
H TT
T sinh kD
(4.155 )
w
and thus the parameter of importance in predicting a concen
tration C(z ) in the vicinity of the bottom is the ratio of
a ^
bottom horizontal water particle velocity to sediment par
ticle fall velocity
IT
C(z ) = function ■
a
H
T sinh kD
w
(4.156)
In shallow water, sinh kD Z kD and the above expression can
be reduced to
C(z ) = function
3.
w ^D
s
Note that in equation 4.15 6 the parameter
H
(4.157)
heuristically
arrived at previously, appears.
146
To obtain the relationship of e to wave parameters a
s _^
look at the basic physical mixing phenomena is again neces
sary. The mean product V' C can be correlated and the
ST
degree of correlation expressed by a correlation coefficient
3^ which can be defined as
3 = 2 (4.158)
/ ,/
9 7
C ' V
P
Then analogous to turbulent flow theory (54) the flue '
tuatmg concentration y C ' can be related to the mean concen
tration gradient as
C'^ = (4.159)
where 5^ = a mixing length analogous to the mixing length of
Prandtl (64) for turbulent flow. Using the above two equa
tions a new equation for the turbulent fluctuation can be
found as
=  ISi! V'y^ • h§ (4.160)
where the minus sign expresses the fact that the transport is
in the direction of decreasing concentration.
The product ^^\J ^ ^'^ is known as diffusion coefficient
for sediment and is given the symbol e^.
147
where l^ is in length units, is dimensionless , and ]J V^^ is
the root mean square fluctuating velocity component.
It will now be assumed that in spilling breakers the
turbulence level is constant over depth and that the mixing
velocity scale is proportional to the turbulence velocity
which, in turn, is proportional to the average energy dissi
pation in the water column averaged over depth.
I— (Ec cosa)
 ^ ^ (4.162)
Using shallow water assumptions, linear wave theory, and
assuming as before that wave angles are small the following
equation can be found
Now, the mixing length l^ will be assumed proportional
to H which is the vertical distance a water particle on the
surface travels through during the passage of a wave.
H^ " E (4.164)
The sediment mixing parameter can then be found from
equations 4.151, 4.16 3 and 4.164 to be (with = constant)
£ = constant ^ ;g '^^^^ (4.165)
148
The data to be used in the assessment of the parameters
C(0) and e come from field studies done by Fairchild (119)
and Kana (120). In the study of Fairchild, suspended sedi
ment concentrations were taken throughout the surf zone from
a pier at Nag's Head, North Carolina. The concentrations
were measured using a tractor mounted pumping device attached
to a pipe manifold extended over the pier into the surf
ione. Time average concentratioixs were taken and the concen
tration was measured as the dry weight of suspended sediment
filtered out per weight of fluid pumped.
In the measurements of Kana, the concentrations were
obtained at a location on the beach near Price Inlet, South
Carolina by individuals stationed throughout the surf zone
using water samplers activated by the individuals. The
devices are described in (120). The concentration is then
the weight of suspended sediment per weight of water sample.
Data used consisted of depthconcentration profiles for
spilling breaking waves in the surf zone. Wave parameters
were measured by a wave gage at the end of the pier in the
case of the Fairchild data, and by visual wave estimates
(using wave poles and stopwatches) in the case of the Kana
data. Unfortunately beach slope at the data collection
sites was not taken. In most all cases the data gave reason
able results with the assumption that the mixing parameter
is constant throughout depth, that is, the data plotted up
149
as straight lines on a semilogrithmic plot of concentration
versus elevation above the bottom. Data sources and data are
given in Appendix B. The data used were for spilling break
ers although in a few cases it appeared that the concentra
tions were scattered with depth (not straight line) in some
of the questionable type breaking wave data. In these cases
it was assumed that the data scatter was due to plunging
type breakers and the data were discarded .
From the data plots and the sediment property analysis,
values of were calculated and values of C(0) obtained
from plots. These values were then plotted as functions of
the relationships provided by equations 4.157 and 4.165.
The results of the analysis are shown in Figures 4.9
and 4.10. Figure 4.9 shows that in fact the sediment mixing
coefficient is well represented by a linear trend with the
parameter  — g — which is a measure of the turbulent
energy dissipated throughout the water column modified by a
mixing length Cin this case wave height). Figure 4.10
provides a functional relationship between C(0) and a par
ticle lift to weight ratio parameter. At lower values of
the lift to weight ratio parameter the concentration C(0)
drops off sharply while at higher values there appears to be
an approach to a linear trend. More data would be needed to
confirm this trend outside the limits of the existing data.
The bed load efficiency factor is found from argu
ments postulated by Bagnold (45) concerning a movable
150
151
+
S a. Q
i£2 X $
o
c
o
•r
+>
fO
S
s
(>
0)
+J
OJ
O)
u
5=
c
(13
o
S
o
ns
a.
u
o
OJ
■r—
P
•r
o
q:
00
<u
0)
+j
OJ
o
CQ
Q+>
•r—
<+
J=
_i
O T3
•r—
+J
n3
^ o
ro ^
•1—
Li
eu
152
boundary. Bagnold (US) postiilates a continuous moving grain
carpet relative to a stationary bed. Assuming a velocity of
the moving carpet = the shear on the carpet (assuming
2
fully turbulent flow) is t ~ (VV^) . The transporting work
2
done is then xV " V CVV ) . This function has a maximum
c c c
value when = ^ hence the maximum transport efficiency is
TV ,
c _ 1_
®c " tV 3"
Bagnold then postulates a similar slip of the actual
grains within the moving grain carpet relative to the carpet
which introduces an efficiency factor e = jj— such that
g
e
e, = e • e = Again the work rate on the moving grains
b c g 3 ^
n '
has a value V, (V V, ) where n' varies between 1 and 2
b c b
according to the local grain Reynolds number (V^Vj^)d/v,
being 1 in the Stokes law region and 2 for large grains .
As before the work rate has a maximum when V, /V =
b c
l/(n'+l). The exponent n' for a given grain size d and slip
velocity of grains V^V^ can be obtained from the slope of
the experimental log curve of the sphere drag coefficient
versus the local grain Reynold's number. Values of e and
e^ can then be found corresponding to a mean flow velocity V
for a given grain size. As per Bagnold (45), the values of
e^ and e^ are given in Figure M . 11 over an extended range of
values of V.
153
The dynamic Bedlload friction factor tan^) has been
mentioned previously to be dependent on solid/fluid param
eter analogous to a fluids Reynolds number. Bagnold (121)
found that tan<i> was dependent solely on a parameter
d /v^
~ where as before, X = ratio of the mean diameter
^ ^ ^ X
of the solids to the mean free distance between the solids
and p  denslty of the solids.
The parameter R^ is thus analogous to a "shear" Reynolds
number of the form
^ _ d /t/p _ du^
shear ~ y / p " v
In experiments using grains of equal density to the
fluid being sheared, Bagnold (121) measured normal and
tangential stresses due to the grain collisions in the
annular space between two concentric cylinders as a function
of the parameter R . The results of this series of tests
s
are given in a plot by Bagnold (121, 45) and reproduced here
as Figure 4.12. The value assumed for X was equal to 14
which was the limiting concentration at which a sheared
array of solid grains was found to cease to behave as a
Newtonian fluid and to begin to behave as a rheological
"paste" (121).
154
o d o d
155
CVJ
T3
CM
o
8
<M
O
O
g
o
CO
O
d
d
a
o
UDj
rO
d
156
4.4.5. Model Comparison with Laboratory and Field Results
In a comparison of the model with existing sets of data
for sand transport, four data sets were chosen, two sets of
laboratory data and two sets of field data. Although there
are many existing data sets as discussed in Chapter II,
other data sets failed to include vital wave parameters
necessary to the model comparison; most typically, breaking
wave, .ajigla was^nat given but ojily the offshore wave angle
measured at the wave generator (in the case of laboratory
data) or in deep water (in the case of field data). It was
felt that transforming wave parameters to breaking condi
tions involved inclusion of an additional uncertainty factor
in the data and was therefore undesirable from the stand
point of assessing model validity.
The four data sets chosen for model assessment include:
the Shay and Johnson tests (laboratory); the Fairchild tests
(laboratory); the Watts tests (field); and the Komar tests
(field). Parameters were calculated in accordance with
equations, graphs, etc., of this chapter and the calculated
model parameters are listed in Appendix B. Wave heights are
assumed as root mean square wave heights in the case of all
data. In the Watts data, model parameter values were
averaged on the basis of 0.5 to M days of wave records, the
time interval in which the sand transport values were given.
As was shown earlier, the fact that the longshore
current is not linear across the surf zone causes the value
157
of in equation 4.126 to be different than the postulated
value.
Also, due to the randomness of waves in nature, it
might be expected that a considerable portion of the break
ing waves might be of a plunging type though the average
wave conditions are of a spilling breaker type. Kana (12 0)
has noted considerably higher concentrations of sediment in
the plunging typ^ breaJcers which would predict much higher
suspended load and, in turn, higher total sand transport.
For these reasons, the sand transport model correlation was
made plotting the immersed weight sand transport I,^ with the
factor XP^
The plot of all four sets of data which cover four
orders of magnitude of the 1,^^ range is given in Figure 4.13
by the best fit equation:
= 2.82xPj^ . (4.166)
A more detailed plot of the Fairchild data alone is shown in
Figure 4.14 where the best fit equation to Fairchild data is
given as :
= S.OlxP^^
The best fit equation to the field data of Komar and Watts
is given by the equation:
I, = 4.24XP,
and is shown in Figure 4.15.
158
Figure 4.13. Sand Transport Model Relationship
159
I^(#/sec)
Figure 4.14. Best Fit Sand Transport Model for Fairchild Data
160
Figure 4.15. Best Fit Sand Transport Model for Field Data
IGl
As assumptions .liav.e ieen incorporated into the model
which cannot be checked out by the present data (i.e.,
roughness, spilling breaker assumptions, monochromatic wave
assumption) the model appears to provide a reasonable fit.
Additional supporting evidence for the bed load component of
the sand transport model is given in the work of Barcelo '
(82) who ran tests as noted in Chapter II in which he noted
that no component of suspended load was present below a
critical wave energy level. For wave energy levels below
that critical value (bed load), the sand transport was found
to be proportional to the square of the sine of the breaking
wave angle x 2 , that is :
Q. ~ sin^(2a )
Conclusive proof of the postulated model must await detailed
laboratory testings in large wave basins with measurements
made of all necessary wave, beach, and sediment parameters.
CHAPTER V
APPLICATION OF SAND TRANSPORT EQUATION
TO CALCULATION OF LITTORAL DRIFT USING SHIP WAVE DATA
5 . 1 Application of Model
In this chapter the equation derived for sand transport
will be applied to compute littoral drift using a source of
visual wave observations made from ships.
Prior to application of the sand transport equation it
is worthwhile to review some of the important assumptions
inherent in the method to be used.
CD Linear theory is valid for the wave transformation
process and the wave energy present in the wave
system;
(2) Assumption in calculation of a "friction"
coefficient are not violated Csee Reference 59);
C3) Bottom topography is composed of straight and
parallel bottom contours;
CU) No drastic changes in the bottom profile are
encountered in the shallow areas seaward of the
breaker line up to the beach;
(5) Adequate sources of sand are available for trans
port;
Item (1) refers to the mathematical formulation of the
problem and its relation to physical reality. As has been
162
163
previously noted, this assumption is reasonably good up to
the region of breaking waves where it departs from the
actual situation. Item (2) assumptions will be discussed
later. Assumption (3) is necessary for the simple applica
tion of Snell's Law of Refraction used in this report and
does not require a monotonic decrease in depth toward shore,
but only the aforementioned relationship between bottom
contours. Assumption (4) is necessary due to the use of
offshore wave conditions for the computation of longshore
energy rather than nearshore conditions. Thus, rock or
coral reef might cause a large dissipation or reflection of
energy before the wave reaches the computed breaker zone,
which would be incorrectly included in the estimation of
expended wave energy in the surf zone. An additional assump
tion inherent in the same transport model is Item (5), the
availability of sand to be moved This is dependent on the
geologic processes acting in the area, and the natural or
manmade conditions present. In some shoreline areas of
Florida there is a lack of sand, predominantly in areas
having extremely low wave energy. Rivers, inlets, jetties,
groins, seawalls, prominent headlands, and submarine ridges
and valleys can also cause a lack of sand in an area downdrift
of an obstacle. A lack of sand supply causes erosion and in
turn a depleted sand reservoir, with less sand available for
the transport downdrift of a barrier.
1614
The littoral drift , can be e:xpressed in terms of a
volume transport rate rather than immersed weight transport
as noted in Chapter II by the following conversion
I, = (p^p) g(lp^) Q, (5.1)
Due to the present more popular method of expressing the
transport rate as a volumetric rate, the results of this
chapter will provide values of (volumetric transport
rate) in cubic yards per day.
5^ 1:1 r Pata "Source
The wave data used in the computation of longshore
energy flux and consequent littoral drift in this report can
be found in the U.S. Naval Weather Service Command — Suimnary
of Synoptic Meterological Observations , Volumes 4 and 5
(122), hereafter referred to as SSMO. These volumes are a
compilation of meterological and sea state observations
taken from ships travelling through "Data Squares" defined
by their latitude and longitude boundaries. The percent
frequency of wind direction versus sea heights can be found
in SSMO Table 18 for different data squares on a monthly and
annual basis. The percent frequency of wave height versus
wave period for both sea and swell observations can be found
in SSMO Table 19 for different squares on a monthly and
annual basis. Computations of sand transport will use the
data from both of these tables. Necessary assumptions made
in the use of SSMO data are presented and discussed below.
165
In the use of Table. 18 the., assumptions have been made
that (1) swell waves are in the same direction as the sea
waves, which in turn correspond to the wind direction; and
(2) waves are propagating in one direction only, the ob
served direction, in any specific time interval. In apply
int Table 19, the assumptions are made that (1) sea and
swell waves of the same period and height can be treated
alike, and will not lose energy to the atmosphere between
the point of observation and the portion of coastline con
sidered; (2) no other wave heights or periods af^e present
during the observation of recorded wave with a given height
and period; and (3) all observations were made in" "deep
water" (h>_2.56T^ in ft.) for the wave periods recorded.
Correlation between the ranges of wave heights, peri
ods, and directions given in the SSMO data volumes and the
corresponding values used in the calculations of drift can
be found in Appendix II. Due to the nature of human obser
vation of waves, the heights and periods found in the data
tables should be considered assignificant heights and
periods .
5.1.2 Analysis of Wave Data to Compute Sand Transport
The immersed weight sand transport rate as given by
Equation M.12 7 can be modified to provide the volumetric
sand transport by the equation 5.1 which can then be ex
pressed as
166
YCS 1)(1p )
(5.2)
where the terms x and depend on wave climate parameters
H^, T, as well as sediment fall velocity. The above
equation represents sand transport in terms of one wave
height, period, and direction, in a deterministic sense.
Considering a continuously changing state of offshore
wave .conditions, heights, .periods, ..and directions , the total
littoral sand transport would consist of average sand trans
port weighted in accordance with values of for represen
tative wave heights, periods and directions. Thus for
continuously changing wave conditions , the total longshore
sand transport as averaged over a time interval t* would be
^ ^ Jt=o Jt=o ^
The value ^ can be thought of as the fraction of time over
which a specific wave having a certain height, period, and
direction is being generated during the period t". Express
ing these results in finite intervals :
f(H^,T,9) = frequency = ^ (5.4)
and
t=t*
Q = I Q/t).f(H^,T,e) (5.5)
t=0
1B7
where
F =00
O T = oo e = 2TT ^
I I I f(H^,T,0) = 1.00 C5.6)
H =0 T=0 9=0
o
with 6 equal to the azimuth of the direction from which the
wave is propagating. It is related to by the equation:
a = 0  9 where 9 is the azimuth of the perpendicular to
on n
the shoreline (see Figure 5.1).
For waves reaching the coast, the summation would be as
follows with 9 = 9^  ct^ and ranging from 90° to +90°;
H „ 9=9 +5
0 T=<=° n 2 ^
1 I I ^ f(H T,9)<1.00 (5.7)
H=0 T=0 9=94
o n 2
Note that in the above summation, when waves are being propa
gated away from the coast, that no wave energy will be avail
able for sand transport. Therefore the total sand transport
becomes
Qo = I I l"" ^ Q (H ,T,9) . f(H T,9) (5.8)
^ H =0 T=0 9=9 1 ^ ° °
o n 2
The value of f(H^,T,9) can be computed by means of SSMO
Tables 18 and 19. From Table 19 a value of f^g(H^,T) is ob
tained such that
H ="
^ig^H^'T) = 1.00 (5.9)
H =0 T=0
o
From Table 18 a value f,o(H ,9) can be obtained corresponding
X o o
to a wave height range in Table 19 such that
158
Figure 5.1. Definition of Azimuth Angle Monnal to Shore e^,
and Azimuth Angle of Wave Propagation e
169
f,o(H*,e) = 1.00 (5.10)
6=0 1^ °
where the * represents the correspondence of in Table 18
to the same range in Table 19 . Multiplying these two fac
tors together gives the desired frequency as a function of
wave height, period, and direction.
?(H^,T,e)  ?^g.?^g (5.11)
By the use of Equation 5.8, the longshore sand transport
can be obtained in cubic yards per day, as averaged over any
given period of wave observations. As mentioned previously,
the representative values of H^jTjB for the ranges given in
SSMO are discussed in Appendix C.
The procedure for the calculation of longshore sand trans
port Q,(H ,T,a )«f(H ,T,9) is as follows:
X. o o o
(1) Compute the onshore directed component of energy
flux
' o
c cos a.
from deep water conditions, that
8 go o
is, the representative conditions for given wave
height, period, and direction ranges (where
E = — ^)
^o 8 ^•
2
(2) Compute the quantity , a bottom frxction energy
dissipation coefficient discussed in Chapter II, to
a shallow water depth, h , outside the zone of
170
breaking waves by numerical integration procedure
of reference (59) (along the coast of Florida this
depth was normally taken as 10 feet).
(3) Calculate the breaking wave angle by an equation
based on Snell's Law and linear wave theory as
postulated by LeMehaute and Koh (12 3)
a, = a • (0.25 + 5.5 E/h) (5.12)
bo o o
 my Calculate the co efficxHut "x and the longshore
energy flux P^^ (as modified by friction dissipation
(59) where
2 2
P = Kj:,(E, c ,cosa, )sina, = K^(E_c _cosa^)sina,
Jifbgb b b fogo o b
(5.13)
(5) Find f^g and f^g values in SSMO Tables 18 and 19
as mentioned previously, and calculate f=f^g'f^g.
Calculation of is then a simple summation process
in which the data must be put through a "filter" to elimi
nate all differential bits of sand transport with azimuth
directions 0 that are less than 9 0° or greater than +9 0° to
the coastline azimuth 6^. When looking offshore a positive
values of = Qi^^^ is recorded for waves propagating from
the left side and causing longshore sand transport to the
right; and likewise, a negative value of Q^^ = is re
corded for waves propagating from the right and causing
longshore sand transport to the left (see Figure 5.2).
Summing the positive, negative, and total values of long
shore sand transport gives the quantities Qjj^+j ^Jl' "^ilnet *
171
Figure 5.2. Relationship Between Direction of Wave Propagation
and Direction of Longshore Sand Transport
172
Additional assumptions used in the preceding method of
calculation which were not previously discussed are:
(1) There is minor loss of energy through bottom fric
tion between h^, the shallow water depth at which
is calculated and the breaking depth.
(2) Computation of K^, , and involve linear
theory and inherent assumptions.
(3) i_s ca.lculated usiQg a bottom profile perpendicu
lar to the stretch of shoreline considered rather
than the actual profile over which the waves
travel. Inherent in this procedure is an addi
tional assumption that the wave climate used
occurs at a point offshore perpendicular to the
portion of coastline considered.
(M) Data weighting of wave heights, periods, and
directions is accomplished linearly using the
centroid of the SSMO data square as the source of
the offshore wave data (see Figure 5.3 for location
of SSMO data squares).
(5) Assumed friction coefficient f = 0.02.
w
5 . 2 Results of Sand Transport Computations
Littoral drift "roses" with annually averaged values of
littoral drift in cubic yards per day have been computed
using the SSMO annual data summary tables along sections of
Florida's sandy shores. These are presented in Figures CI
173
174
through C26. Because of the large number of these figures
they are located with the Appendices Section of this thesis.
A littoral drift rose diagram for each section of coast
considered gives the annually averaged total positive and
negative littoral drift. Positive values of littoral drift
refer to drift moving toward the right when looking offshore,
and conversely, negative values of drift are quantities of
drift moving to the. J. eft a^, noted previously. On the East
Coast of Florida a positive value of drift would thus repre
sent Southward drift, while on the Gulf Coast, the reverse
would be true; that is, a negative value of drift would repre
sent Southward drift on the Gulf Coast. The net drift values
represent the difference between the Southward and Northward
total values of drift with the direction of the drift indi
cated by its sign as described above. Although the littoral
drift has been computed for coastline orientations ranging
over 360° of the compass, in actuality, the coastline orien
tations range at most over 18 0° for any given section and
have been presented showing the maximum practical range plus
or minus 20° for local anomalies.
As mentioned previously, these values of littoral drift
are for stretches of coast exposed to the ocean wave climate
as represented by SSMO data. They are not valid for bays,
lagoons, or estuaries, where the shoreline is not exposed to
a wave climate represented by the SSMO data. Also, they are
not valid where local anomalies exist in offshore bathymetry.
175
5.2.1 Use of a Littoral Drift Rose
Use of a littoral drift rose is as follows:
(1) Determine the orientation of coastline at
which a drift is desired.
(2) Using the azimuth of the seaward directed normal
to the coastline at the location, find the values
of total positive and total negative littoral
, ..drift assoaia.ted with thi.^ azimuth angle on the
proper drift rose corresponding to the desired
location .
(3) Find the value of net drift as the difference
between the positive and negative drift values.
If the net drift value is positive, the net drift
will be to the right when looking offshore; if
negative, the net drift will be to the left.
To demonstrate the method, values of net drift at Ponte
Vedra Beach, south of Jacksonville are found from Figures
5.4 and 5.5. The azimuth angle of the perpendicular to the
shoreline is 76°30' as shown in Figure 5.4. Thus, the
total Southward drift is 1600 cubic yards per day, and the
total Northward drift is 810 cubic yards per day from Figure
5.5. The net littoral drift is 790 cubic yards per day or
288,000 cubic yards per year to the South.
Limitations in the simple procedure for calculating
drift values in the above manner will be discussed, taking
into account some of the data limitations and data bias.
176
Figure 5.4. Azimuth of Normal to Shoreline at
Ponte Vedra Beach, Florida
177
Annudlly Av*roij9d Totol Littoral Onff
St. John's Rivor to St. Augusiltie lnl«t
N
Figure 5.5. Determination of Total Positive and Total
Negative Littoral Sand Transport
178
5.2*2 Possible Sources of Data EvjpQr .Qr ^a s
In the SSMO data, possible sources of error include:
(1) Human error and bias in the observation and
recording of the wave data.
(2) Absence of extreme wave conditions due to
routing of ships out of bad weather.
(3) Inaccuracies introduced due to the lack of
swell direction data,
(4) Inadequate resolution of wave data direction wise.
(5) Inaccurate wave height recording due to wave
observation in a strong ocean current.
Error sources (1) and (2) are selfexplanatory. In
regard to (1), it has been shown that a large bias is intro
duced in the directional data due to the observer tendency
toward recording of wave directions along the four cardinal
and four intercardinal .points of the compass. This effect
is seen in the littoral drift roses. It is felt that the
bias should not significantly affect the results presented
here though, since wave directions used in the computations
were reduced to the eight points of the compass in the SSMO
volumes. If it is assumed that the waves were recorded to
the nearest point of the compass (on an eight point system) ,
the maximum error between a recorded wave direction and its
true direction would be 221/2°. It is recommended that
values of drift in a range of azimuth angles 111/4° to
179
i:he actual coastline aziinuth i>e ■TsonsadeTed as the range of
possible drift values, thus covering a 221/2° range of
possible wave directional error.
The original method of reducing the data from 36 points
of the compass to 8 points of the compass given in the SSMO
volumes introduced a skew of the data by an angle of 10
degrees clockwise. This has been compensated for in the
littoral drift roses and offshore wave climate roses by
shifting rose azimuth angles 10 degrees counterclockwise.
As mentioned earlier, the lack of swell direction data,
and distinction between sea and swell, cause the assump
tion to be made that swell waves are being propagated in
the same direction as the local wind waves (which is the
recorded wind direction) . It is unlikely that swell is
always in the same direction as the local seas and this
could lead to considerable error in the computation of long
shore sand transport. In regard to Item ("4), since long
shore sand transport is dependent on wave direction due
to refraction process, the method of computing sand trans
port by using only eight points of the compass poses a
question as to the magnitude of error possible in the
results. It can be shown that the maximum error introduced
by this approach as compared to spreading the energy
evenly over all directions within an octant is 10 percent.
180
In regard to Item .( 5) ,..wave iaights are affected by
strong currents and have the tendency to steepen when
propagating against an opposing current and are reduced
in height by a following current. This effect is noted on
the Southeast coast of Florida where the Gulf Stream is
very close to shore. Due to the fact that shipping lanes
run through and along the Gulf Stream, it is felt that many
of the ob&erved waves approaching shore have recorded
wave heights higher or lower than would be experienced on
the shoreward side of the Gulf Stream in comparitively
still water. This effect would cause the computed South
ward drift values to be higher than the actual drift values
and Northward drift values to be lower.
5.2.3 Other Possible Errors or Bias in Analysis
A possible large source of error comes from the
assumption mentioned earlier that waves are considered to
be propagating in one direction at a time. That is, it is
assumed that when waves are moving away from the coast,
there are no waves reaching the coast, and thus there is no
sand transport at the coast. This is a questionable assump
tion since waves are known to propagate in many directions
at the same time but analysis of such error is impossible.
Other possible sources of error involve the computation
of the friction coefficient and the violation of Snell's
Law with regard to the bottom contours .
181
.The modification of wave height due. to friction effects
as the wave propagates across the continental shelf has
factors which could contribute to inaccuracies as follows;
(1) Friction coefficient
(2) Method of taking profile for a coastline section
(3) Neglecting friction beyond a certain depth, = h
The friction coefficient used in this study was constant,
■equal to 0.02, but is known to be a function of bottom
roughness (as noted in Chapter II), which in turn depends on
wave height, and water depth. Thus, friction is not constant
but varies with time. A sensitivity test was done using
three friction factors: 0.01, 0.02, and 0.0 3 for the location
of coastline which best represents an average profile from
Fort Pierce Inlet to St. Lucie Inlet to compute values of
drift. Assuming that the friction factor 0.02 is correct, a
value of f=.03 gives drift values approximately 20% lower
and a value of f=.01 gives drift values approximately 22%
higher. The sensitivity would be much greater on a broader
shelf width as in North Florida on the East Coast, and much
smaller on a narrow shelf width as encountered in the south
ern limits of Florida on the East Coast.
The method of taking a profile perpendicular to the
stretch of shoreline considered leads to high values
which would tend to overestimate the wave height and long
shore sand transport since refraction effectively causes
182
waves to travel over a longer profile than the one used.
In view of the fact that locations of the individual wave
data observations are unknown, the method of using a profile
along the perpendicular to shore seems a reasonable
approximation though.
Assumption (3) which was mentioned earlier is made
because the value of is based on linear wave theory,
and,, near breaking conditions in shallow water, the wave
form no longer corresponds to linear theory. In addition,
the beach is in a dynamic state at shallow depths which
would make assumptions regarding bottom profile in this
region invalid during part of the year.
5.2.4 Other Potential Sources of Error
Other factors which certainly have a bearing on
littoral drift in an area but which were not accounted
for in the present computations include :
(1) Wind, tidal, and inlet refraction effects on
littoral currents and corresponding drift.
■ (2) Sheltering effects of reefs, rock outcroppings ,
large submerged sand ridges, etc.
(3) Interference in littoral regime due to jetties,
inlets, rivers, sand sources, sand sinks, etc.
Factor (1) has been found to be of major significance
in some studies (12'4).
183
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185
The sheltering effect of reefs and rock outcroppings is
certainly a factor affecting littoral drift along the south
east coast of Florida. Many rock outcroppings and reefs
exist in the littoral regime and definitely influence drift
values. In places such as Cape Kennedy where a large
underwater sand ridge exists, the drift pattern is altered
by the sheltering effect of the ridge which prevents some
northeasterly wav^. from reaching the .southern shore and
some southeasterly waves from reaching the northern shore;
thus, to an extent, the ridge tends to be a selfperpetuating
littoral barrier.
Jetties, inlets, rivers, submarine valley, etc. all
influence the pattern of drift to alter it from the idealized
model used to compute values of drift; these influences must
be recognized when applying drift values derived by the
approach presented.
5 . 3 Comparison of Calculated' Littoral' Drift Rates
W3.th Previously Estimated Values
Comparisons of the present study results with estimated
values of net drift compiled by the U.S. Army Corps of
Engineers are summarized in Table 5.1. The Corps of Engi
neers values were determined by various methods which include
analysis of dredging records, volumetric surveys, and pumping
records at existing bypass plants. Computed values of
drift by the present method provide both an "expected" value
186
of drift and, to illustrate the sensitivity of drift to
coastline orientation, a range of drift values which encom
pass +11 1/4° span of azimuths to the actual coastline
azimuth, 8^, at a given location. Total positive and nega
tive drift rates with corresponding ranges of values are
summarized in Table 5.2.
Note that some of the comparisons presented in Table
5.1 and Table 5.2 may be misleading in that they are ex
tremely close to Corps estimated values when the assumptions
involved in the program to compute drift are possibly vio
lated. One such location is Fort Myers Beach on the Gulf
Coast. The computed value of net drift is 21,900 cubic
yards per year in a Northerly direction which is extremely
close to the Corps estimated value of 22,000 cubic yards per
year. The assumption of parallel offshore contours is
violated here though, and, refraction of waves from a North
easterly direction is undoubtedly much different from what
the simplified analysis based on Snells ' Law would compute
it to be. Refraction of N W Waves off Sanibel Island would
tend to create a complex nearshore current situation with
the probable direction of drift being North even if the wave
climate and ideal bottom topography would normally tend to
create a Southerly drift.
It is suggested that use of the results presented in
the littoral drift roses be carried out with a knowledge of
187
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189
the assumptions present in the'study such that one is
not misled by the seemingly good comparisons as given
above which may be fortuitous .
Along the Atlantic Coast, the SSMO data confirm the
Corps estimates of net Southerly movement of sand, and
on most of the lower Gulf Coast the data confirm net sand
movements in a Southerly direction. Along the Panhandle
the net movement is in a Westerly direction which also
agrees with other studies. Except for certain anom.alies
in drift directions due to coastline orientation, the over
all trends confirm past observations with regard to direction,
although magnitudes are different. Reason for the extremely
high values of drift computed in Southeast Florida are not
known at this time although the author speculates three
possibilities :
(1) The effect of the Gulf Stream current
on wave height observations as mentioned
earlier.
(2) Effects of the Bahama Banks.
(3) Resolution of wave data into large data
squares rather than smaller squares where
overall offshore conditions are the same.
An interesting observation was made in this study with
regard to null points in net drift. By viewing either the
net drift diagrams or the total positive and total negative
190
drift diagrams, it can be seen that two types of null points
exists in the drift regime. In Figure 5.6(a) a "Type 1"
null point is shown for a portion of a typical total drift
diagram. Assume first that an island exists such that its
original orientation conforms to the null drift point (total
positive drift = total negative drift), Figure 5.6(b). A
perturbation in the system such as a storm, or the building
of jetties ^at ends xsf.. the. .i^lan.d could cause the sand to be
shifted to a position shown in Figure 5.6(c). In this case
the net drift on the right side of the island would now be
to the right while the net drift on the left side of the
island would be to the left. Thus the overall effect of the
perturbation would produce instability in the island, with
the net result that the perturbation would increase and
eventually the island would experience a breakthrough as
shown in Figure 5.6.
The orientation of the Gulf Coast shoreline in Lee
County, Florida, is found to be approximately characterized
by Type 1 null point. It is noted that this section of
coastline contains numerous inlets and has a history of
inlet breakthroughs. Another area where this type of null
point is experienced is the Gulf Coast near St. Petersburg.
Islands in this region tend to be extremely concave and
would have probably broken through by now if not for the
extensive groin fields hindering the transport of drift in
191
ji"^ (b) Ideal Island oriented
1o null point zero
net drift
Perturbation in system
causes orientation of island
with associated drift pattern
(d) Instability leads to eventual
breaitthrough
Figure 5.6. Ideal Case of an Unstable Null Point
192
the region. It should be noted that a perturbation in the
convex sense would also be unstable and lead to an increasing
convexity. No cases of this type system were noted on the
open coast but numerous such features are seen in bays due
to this type drift system.
The second type of null point is shown in Figure 5.7(a).
An ideal island when oriented to this type of null point has
a tejidency to sta bilxze... atae.l.f once ^ perturbation in the
system drives it from the ideal state. Figures 5.7(b),
5.7(c), and 5.7(d) show the series of events leading to
stability. Part of the East Coast of Florida is near this
type of null point where a predominant tendency for few
inlets exists. Many of the inlets (such as Sebastian Inlet
which occurs very near a "Type 2" null point) have had a
record of numerous closures after being cut. Of course,
many additional factors, influence stability and instability
in true physical systems such as the amount of drift supplied
to an area, and the ocean tidal ranges. These additional
effects may overshadow those discussed here. It is hoped
that in the future this theory can be explored further.
Some further insight into the "null point" type can be
given by considering two different cases of offshore wave
climate, a unidirectional wave climate (all waves from one
direction) and a bimodal wave climate (waves coming from 2
directions, 18 0 degrees separate). For simplification the
193
(a) Type H null point
3n
(b) Ideal Island oriented
to null point — zero
net drift
(c5 Perturbation in system couses
orianlolion of Island witli
associoted drift pattern
id) Selfstobiiizing drift pattern
leads to original isiond
conditions
Figure 5.7.' Ideal Case of a Stable Null Point
194
sand transport wi 1 1 .all .be .considered to .be of suspended
load and the sand transport equation to be of the simplified
form
= Ep sin2a^ (5.14)
where
 — * W( S 1) (1p "7 constant (5.15)
g ^o ■
Now when waves are coming from a predominant direction as
shown in Figure 5.8, a littoral drift rose can be postulated
as shown in Figure 5.8 where it has been assumed for simplic
ity that Ep=1.0. As the littoral drift rose drawn in Figure
5.8 is a "net" littoral drift rose it is apparent that this
type of drift rose applies to the type II shoreline, a
stable shoreline.
When waves are coming from the directions as shown in
Figure 5.9, and each directional climate contains the same
amount of energy, a "littoral drift rose" can be postulated
as shown in Figure 5.9. This drift rose can be seen to be
of the unstable type of drift rose which leads to pertuba
tions in the shoreline. This type of wave climate and drift
rose is believed to be the cause of the large pertubations
(like small capes) in the shorelines of many elongated bays.
An example of such a bay is Santa Rosa Sound in the Panhandle
of Florida where a number of such anomalous shoreline "capes"
exist (see Figure 5.10). The types of wave climate expected
195
WAVE CLIMATE CONSISTS OF
ONE DOMINANT WAVE DIRECTION
i
Q^^ ECg sin 20
^ y> y y y y
Figure 5.8.
Stable Type Littoral Drift Rose
Due to Unimodal Wave Climate
196
Q^/v Ecg sin 20
WAVE CUMATE Bl MODAL
FftRALLEL TD SHORELINE
FOR AZMUTH SHOWN
Figure 5.9.
Unstable Type Littoral Drift Rose
Due to Bimodal Type Wave Climate
197
198
in these elongated bodies of water are conducive to "littoral
drift roses" of the unstable type where a long fetch along
the axis of the bay or sound allows large wave energy compo
nents coming from large angles while little wave energy
comes from directly across the sound due to a restricted
fetch. This is the type of wave climate conceptualized in
Figure 5.9. Previously no explanations for these shoreline
..shapes existed.
5 . 4 Comparison of Estimated and Observed Wave Climates
To determine the reliability of the SSMO data and
computed shoaling, refraction, etc. effects in the present
study, a comparison was made using wave records obtained
from shorebased gages. Data from step resistance wave
gages operated by the Coastal Engineering Research Center
were made available for three wave gage stations : Daytona
Beach (East Coast), Lake Worth Palm Beach (East Coast), and
Naples (Gulf Coast). Wave data were obtained intermittently
during the years of operation of these stations due to
various storms damaging equipment or structures on which the
gages were mounted. To avoid a seasonal bias in the shore
based recordings, a sample of data best representing the
average annual conditions was used in each comparison.
Table 5.3 shows the observation period used, the total
number of observations, and the depths of these stations.
199
In regard to the SSMO data, certain assumptions had to
be made with respect to the frequency of occurrence for wave
heights and periods. Only the onshore directed waves were
used for obvious reasons, which gave an extremely high
frequency of "calm" conditions (H=0) at shore. It was
assumed for the plotting of cumulative height curves that
the sea state at shore is best represented by wave heights
of JLess, than one £oot wh.en of f ■shore directed waves were
being recorded. Most likely, many waves greater than one
foot would be recorded at shore during this time. This
assumption gives a poor basis of comparison for recorded and
observed low wave heights in which the majority of waves
fall. In the cumulative distribution curves for wave period,
the assumed frequency of occurrence of a specific wave
period was assumed equal to the frequency of the onshore
directed wave (of a specific period) times one (1.0), divided
by the total fraction of onshore directed waves .
Cumulative curves of the plotted wave height and period
distributions at these three stations are shown in Figures
5.11 through 5.16. The wave height cumulative curves show
three sets of points with corresponding smooth curves drawn
through them, one curve for the CERC gages, one curve for
the deep water onshore wave climate as recorded by SSMO, and
one curve for the SSMO wave climate as modified by the
present study to the depth of the recording wave gage.
200
Table 5.3. Recording Periods of Shore Based CERC Wave Gages Used in
Comparison of Actual to Predicted Shore Wave Climate.
Daytona Beach, Florida
(Depth of Wave Gage = 15 ft. MWL)
FebruaryDecember 1954
FebruaryNovember 1955
FebruaryMarch 1956
January April 1957
NovemberDecember 1954
Lake WorthPalm Beach, Florida
(Depth of Wave Gage = 18.2 ft. MWL at Lake Worth and
= 15.7 ft. MWL at Palm Beach)
January
April and
June
December
1958
Palm
Beach
1161
observations
January
December
1960
Palm
Beach
2020
observations
January
December
1960
Palm
Beach
1687
observations
January
April and
June
December
1961
Palm
Beach
1301
observations
January
December
1966
Lake
Worth
1751
observations
Naples,
Florida
(Depth of Wave Gage
= 16.5 ft.
MWL)
January
December
1958
1454
observations
1570 observations^*
1151 observations
234 observations
321 observations
304 observations
"'Each observation is the significant wave height and period as deter
mined from a 7minute recording of sea surface elevation measured
using a step resistance type wave gage.
201
202
a
o
i
203
as
•a
•r
o
CP)
204
IT3
a
s
o
0)
i
•
205
206
207
Since it is assumed that periods are not modified by offshore
topography, two curves are shown on the wave period cumula
tive distribution curves, one for CERC recorded wave climate,
and one for SSMO recorded wave climate with inherent
assumptions .
The curves show that wave heights of the higher energy
waves are represented well by the modified SSMO data.
Unfortunately , thojjgh, .p^iods are .poorly defined by the data
source. Due to the large dependence of wave modification on
wave periods, it is felt that an even closer correspondence
to offshore observations might be obtained with improved
period observations.
CHAPTER VI
ANALYTICAL MODELS FOR SHORELINE CHANGE
6 . 1 Introductory Remarks
As important as the ability to predict sand transport
is, the problem of being able to predict shoreline change is
equal 1 y i mportant when evaJ.iL3,±ixLg .optimal location of build
ing in the coastalbeach strip. The effects caused by
reefs, rock outcroppings along the beach and coastal struc
tures (revetment, groins, jetties) on the shoreline are
important to coastal engineers . A predictive capability in
determining shoreline changes due to longshore sand trans
port changes becomes of primary importance in the design of
coastal structures also. If accurate predictions cannot be
made for determining the changes due to coastal structures,
then such projects may be doomed to failure.
This chapter first discusses a heuristic model for
determining an "equilibrium" shoreline (a shoreline along
which the longshore sand transport is zero) under given
average wave conditions and is applied to specific areas of
Florida's coast where nearshore reefs play an important role
in determinirtg shoreline shape. The model is then given a
more sound theoretical basis using the longshore sand trans
port equation.
208
209
A second model (dynamic and analytical) is then postu
lated using the sand transport equation and the equation of
continuity via linearizing the sand transport equation
similarly to the work of PelnardConsidere (125) only with
different parameters. Analytical solutions to the partial
differential equation can then be found and solutions for
various coastal structures (e.g. jetties, beach nourish
ments) can.be found and nojid i Tn e n .siona3J.zed solution graphs
given.
6 . 2 Heuristic "Equilibrium" Shoreline Model Development
It has long been recognized (126, 127) that the shore
lines of sandy beach areas protected by headlands follow the
shape of a logarithmic spiral curve. The most prominent
logarithmic shoreline features noted in the literature are
associated with natural rock headlands on what Johnson (128)
termed as "shorelines of submergence." Shorelines sheltered
by natural reefs, shoals, and coastal structures can also
exhibit this logarithmic spiral shape though as shown in
Figures 6.1, 6.2, and 6.3, exemplifying that such natural
shapes are not just commonplace on naturally rocky coasts
but exist on most all shorelines with sedimentary beaches.
Figure 6.1 represents a large shoreline indentation just
south of a natural offshore "worm" rock reef on the east
coast of Florida near Vero Beach where the predominant
direction of sand transport is southward as shown in Chapter
Figure 6.1. Longspiral Curve Fit to
Shoreline at Riomar,
Florida
211
Scale — KilomeJera
212
GULF OF
MEXICO
Depth Contours in fest.
Shoreline and Offshore Depth
Contours from Nautical Chart
N.O.S. 856 S.C.
I I L
0 0.1 Q2 Q3 Q4 05
Scale— Nautical Miles
1 — I I I
0 .1 2 3 4 .5
Scale  Kilometers
Figure 6.2. Logspiral Curve Fit to Shoreline
at Lacosta Island
213
o
o
CD
■o
fSS
Q.
o
>
o
s_
•1—
CL.
to
o
vo
0)
S
2m
V. Figure 6.2 represents a large shoreline indentation on
the lower Gulf Coast of Florida on La Costa Island. This
section of coast is also believed to have a net southward
sand transport as per Chapter V. A nearshore shoal appears
to be the cause of the indentation to the south. Figure 6.3
shows that this phenomena is not limited to only small
sections of shoreline but can be somewhat representative of
a large section of coa^t such as the Panhandle section of
Florida west of Cape San Bias where the predominant direc
tion of sand transport on the beaches is westward (to the
west of Panama City).
As the logarithmic spiral is found to describe smooth
sections of coast so well it might be reasonable to ask if
such a fit is just a quirk of nature or whether perhaps
there is a more justified reason behind the phenomena? Some
insight can be gained by looking at the equation of the
logarithmic spiral which is of the form
R = e^^ (6.1)
The above equation can be reformulated into an equation of
the form:
lnR2  InR^ = k(e2  8^) = kAS (6.2)
or
R^/R^ = e^^®. (6.3)
When A9 is a constant arc size the parameter R2/Rj^ is also
constant for a given value of "k." By a judicious choice of
215
the value "k" and of the use of a given section of the log
spiral curve, an unlimited number of "smooth" curves can be
drawn and fit to curved shoreline features. Figures 6.1
through 5.3 have been plotted using different values of the
parameter R2 / Rj^ .
Sea shells also have naturally occurring smooth curves
and can be found to fit a logarithmic spiral curve quite
closely also (129).,. There is no physical justification for
this shape though, and indeed it is fortuitous, as shell
shape can be found to closely fit more physically reasonable
shapes such as ideal airfoil curves (129).
Justification for the log spiral shoreline shape as
well., appears to be lacking from a physical point of view
and hence a more physically descriptive system for the
equilibrium shoreline shape needs to be considered as in the
following heuristic model.
The present heuristic model describes the shoreline
sheltered by a headland (or on a smaller scale, a groin or
offshore breakwater) using a wave energy diagram consisting
of representative offshore ship wave height and direction
observations. Offshore ship data ane summarized by the U.S.
Naval Weather Service Command in a publication entitled
"Summary of Synoptic Meteorological Observations (SSMO)"
(122), as discussed previously and used for computing sand
transport along the coast.
216
To describe the simplest case of the present heuristic
model. Figure 6.4 is presented with an xy coordinate system
parallel and perpendicular to the existing overall trend of
a large coastline segment. Note that stavting in this
ahapterj the xy coovdinate axis convention has been changed
to aovrespond to simitar notational convention used by othev
authors. The y direction now refers to an axis yerpendi
oular to the. shorelin& .j^hil& the x axis now refers to the
axis parallel to the shoreline . In the simplest case of
equal wave energy flux approaching the coast from all direc
tions the wave energy directional distribution is described
by a semicircle as in the top portion of Figure 6.4. At a
point "P" described by coordinates (x, y) in a sheltered
section of coast, by geometric considerations shown in
Figure 6.4 it can be reasoned out that the slope of the
shoreline in plan view, must be described by the equation:
dy , , $ V
= tan (2") (5.4)
if there is to be no longshore component of wave energy flux
(which has been shown earlier to be proportional to H^'^^)
striking the shoreline in its final equilibrium shape. It
should be noted here that the condition of no longshore sand
transport along the coast is the same condition as that
generally assumed to give the logarithmic spiral shape
shoreline by earlier investigators (126, 127). This is a
static condition since no dynamics of sand transport are
217
Figure 5.4. Orientation of a Shoreline Segment
Having Equal Offshore Wave Energy
218
involved in the shoreline shape. Thus only a final shore
line shape can be arrived at and not intermediate time
steps. Upon expanding the equation in terms of $ it can be
shown that :
g . tan ( tan^ y/x ^ ^5^5)
which can be solved for y in the form
y^ = Constant • xCl + ^1 + (y/x)^) • (6.6)
This equation can be solved by iteration for a shoreline
shape or solved graphically from the original differential
equation as expressed in Equation 6.4.
The more complex case of this simple static equilibrium
model occurs for the typical case of an unequal distribution
of wave energy flux impinging on the coast. For example,
off the east coast of Florida the average wave energy flux
distribution from, the SSMO data is summarized in Figure 6.5.
5/2
This distribution of H is formulated from, tabulated
visual estimates of ship wave observations taken by commer
cial vessels and the military vessels within specified
latitude longitude grid sections in the previously mentioned
publication "SSMO." By summarizing the wave energy from
5/2
given directions using representative wave energy value H
multiplied by the frequency of occurrence of that energy
level, the continuous energy distribution shown can be found
as a function of a direction angle, "Q."
219
Figure 5.5. Wave Energy Rose for Ocean Data Square
#12 Off East Coast of Florida
220
The magnitude of this energy flux level is given by the
value "E*." As the specific units of measurement for this
5/2
wave energy are in (feet) , the given level of energy at a
specific direction angle can be thought of as an (average
5/2
wave height) impinging on the coast from the direction 6
at any time. This is obviously a simplified assumption to
the true situation where wave energy is from different
direritlDxis .at different times o.f the year^and at different
energy levels, but this simple representation will suffice
for the conditions of this model.
Using the energy distribution so described, the total
energy striking the coastline in both parallel and perpen
dicular directions to the x axis in the sheltered area of
the coast can be described by the quantities P" and N"^
respectively where:
•180
vk'^ r ay
cosae
 J E*'2(e)
(6.7)
180
In the sheltered area of the coast the shoreline will
take the shape to minimize the longshore energy component of
the incoming waves, i.e., energy component longshore = 0.
This condition is shown in Figure 6 . 6 and is represented by
the equation
221
Ei_ = Energy Component Longshore =0
=N sinIZ + PcosX2 (in x direction)
Figure 6.6.
Energy Diagram of Coastline with
Equilibrium Coast Shape
22 2
longshore energy compoxLent  0  N"sinfi + P^cosfi (6.8)
or
w = = ai ^6.9)
where the quantity PVN'^ = f(0). Thus the condition of the
shoreline slope at a point in the sheltered area can be
ascertained for the given angle by two integrations of
the offshore wave energy diagram described by equation 6.7.
It should be noted that this model does not take into consid
eration any effects of diffraction and therefore cannot
model the effects of wave action cutting behind a headland,
but where a solid structure such as a seawall, jetty, or
groin is the limiting condition, the model might be expected
to be very realistic.
A graphical solution diagram for the east coast of
Florida where the general shoreline trend is considered to
run directly north and south is presented in Figure 6.7. At
various values of angle $ the slopes have been plotted and
curves fit to the slope values. The curves so drawn re
semble logarithmic spirals quite closely. Looking again at
the same east coast shoreline area which was shown earlier,
in Figure 6.1, a curve from Figure 6.7 has been fit to the
shoreline at Riomar, Florida, in Figure 6.8. As Figure 6.8
shows, the equilibrium shoreline curve fits quite well with
proper adjustment of the xy axis origin.
223
Figure 5.7. Solution Curves to Equilibrium Shoreline
Equation for the East Coast of Florida
224
Figure 6.8. Equilibrium Shoreline Shape for Riomar, Florida
225
Equilibrium coastline curves for a general shoreline
tending northsouth on the Gulf Coast of Florida using a
similar Gulf Coast wave energy distribution is shown in
Figure 6.9. These equilibrium curves also can be approxi
mated closely by log spiral curves. Figures 5.10 represents
the earlier case shown for Lacosta Island, now with the
equilibrium shoreline orientation given. Here it was found
jiecessary to shift the .a,:xj s of the equilibrium diagrams 4 3
48° to get a good shoreline fitting. The necessity of this
shift to provide a good fitting indicates that a shift in
the energy distribution diagram is necessary for the "true"
wave climate, i.e., the predominant wave energy comes more
from the west than from the northwest of the wave energy
distribution diagram.
Figure 6.11 presents an equilibrium shoreline shape for
the Panhandle Gulf coast of Florida using the wave height
distribution from data square number 16 of "SSMO." The
equilibrium shoreline gives an improved fit to the shoreline
shape in this area over the log spiral curve fit previously
presented in Figure 6 . 3 provided a shift in the energy
distribution diagram of 21° is made. This suggests a "true"
wave energy distribution shifted more toward the west than
the distribution obtained from the ship wave observations.
As a final use for this heuristic method of equilibrium
shoreline shape, coastal structures such as offshore breakwaters
226
Figure 6.9. Solution Curves to Equilibrium Shoreline
for Gulf Coast of Florida
227
GULF OF
MEXICO
I L
0 Ql_ 02 Q3 0.4 05
Scnle— Nautical Miies
I — I — I — I— ; I I I I
0 .1 2 .3 4 .5
Scale — Kilometers
Shoreline and Offshore Depth
Contours from Nautical Chart
N.QS. 845 S.C.
Figure 6.10. Equilibrium Shoreline Shape
for Lacosta Island, Florida
228
229
can. be considered to limit the a.va 1 1 able wave energy strik
ing a shoreline. The sheltering effect of the breakwaters
at a point P(x,y) in the lee of the breakwaters is shown in
Figure 6.12 where the wave energy reaching point P is
limited between directions <l>^ and Again the equilibrium
shoreline slopes are described by the equation:
(6.9)
where P* and N* are again calculated by integration of the
wave energy diagram times the cosine and sine of the direc
tion angle 8 respectively. Once the slopes have been cal
culated, smooth curves through the slopes can provide a set
of potential shoreline shapes. In the case that the shore
line has a long stretch of protected coast both updrift and
downdrift (i.e., a large number of offshore breakwaters or
groins are in use) and the potential sand transport is zero
along the coast, the shoreline shape must reach an equilib
rium such that the net accretion seaward of the original
shoreline is equal to the net erosion shoreward of the
original shoreline as in Figure 6.12. From the given shape
postulated, the indentation of the shoreline can be cal
culated and thus the effects of building too close to the
shoreline provided for.
6 . 3 Analytical Treatment of Shoreline Change Model
One can also look at the sand transport equation as
providing a mechanism for determining the "equilibrium
230
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o
c
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o
w
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o
o
Ik.
a
49
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a
a
w
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a
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II
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<s
<u
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+J r
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• i~ o
D. I—
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to
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ui (U
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231
shoreline shape." Once again the sand transport equation is
of the form
e, sin2a, e C, cosa,
D D ^ s D D
tantj) 2 w
s
(6.10)
r _ /V <(lK)tan3N
 ^\ f ^
w
with „
P. = — H c, sina, cosa, as before which is equivalent
it 8 b b b
to the following expression assuming is small and cosa^=1.0
I, = b!' H, ^''^ sin^a, + s! H, ^^^sina, (6.11)
5 1 b bib b
where
 (Y ^(lK)tang s % ,Y jg.
w ^
(5.12)
e c,
s 1
w
w s
Now to proceed, it is necessary to linearize the sand
transport equation in the term sina^ by incorporating some
average value of sinaj,^ into the above equation, i.e.,
i=M
(sina, ) = y f. sina, . (6.13)
b 1 bi
1 = 1
where a multidirectional wave climate is now assumed and f^
= frequency of occurrence of breaking wave with wave direc
tion a, . . Then b" = B"(sina, )" and the sand transport
bi lib
equation for a multidirectional wave climate (time averaged)
is
M
I„ = (b; + S*) I f.H.^^^ sin(9  6, . ) iZ.l^)
% 1 1.^^11 s bi
1 = 1
 (Y i<(lK)tane , ^s'^b ,y Jg.
^1  ^^w f ^ ^8
232
where a, . = 9  8, . = breaking wave angle with shoreline
and 9^^ = defined angle of wave crest at breaking
6 = defined angle of shoreline
s °
This convention is shown in Figure 6.13.
Now for the "equilibrium shoreline" shape which has
been defined as the shape the planform of the shoreline will
take after sand has shifted sufficiently to provide a condi
tion of 1^ = 0 along all parls ryf trhe shoreline, the above
equation ' becomes for a point "P"
I, = 0 = (B* + S*) y f.H.^''^ since  9, .) (5.15)
1 1 1 . 1 a. s bi
2.1
which can be expanded using trigonometric identities and
solved for the angle .8 to equal
M
f.ri. smS, .
tane = ^ = ^ ^ ^ ^ (6.16)
s M
y f .H. ^ cose, .
. ^, i X bi
1 = 1
In the sheltered region of coast the above equation can
be applied at the boundary with a source of wave data and
integrated in a step wise fashion with tan 9 = ^ until the
° ^ s dx
entire planform equilibrium shoreline shape is described.
As the first method is more intuitatively clear (therefore
easier to spot mistakes) the second approach was not needed
but was described for more theoretical insight into the
method only.
233
y
ELEVATION VIEW
Figure 6.13. Definition of Sign Convention for Shoreline Modeling
234
Again it is important to note that the method does not
take into consideration diffraction effects although with a
very detailed analysis diffraction effects could be con
sidered in equation 6.16 by incorporating a diffraction
coefficient K^^ along with the wave height (e.g., H^K^
rather than H^). The diffraction process is discussed in
Reference (130).
HJae previx)us " eqiidJLibrduin shoreline" model took into
account only the static shoreline concept (i.e., where
boundaries must eventually comply with the condition that
there is no sand flux across them) . It is apparent that to
describe the dynamic process of sand transport a time as
well as spacially dependent model must be derived.
Using the total sand transport equation as discussed
previously, one can find the sand transport in terms of
volumetric rate as before
K <(1K) tanB r sin2a, e c, cosa, ^
n  c w , J b b s b b
^l' WcS^1) (lp^)f ^ * \tan4) 2 ^ /
§ o w s
(6 .17)
which by making the small breaking wave angle approximation
2
(cosa^ = 1.0) and linearizing the sin term in the bedload
M
portion of the equation as before with f^sina^^ = (sina^)"
the equation becomes
Q. = F, (B,S ,p ,f ) (— ^ (sina,)' + 251 P. (6.18)
1 ' g^o' w [tantp b w J £
235
The above sand transport equation can be coupled with
the longshore sand transport continuity equation
X=0 (6.19)
which states that the difference in sand flux into a control
volume is balanced by the storage (accretion or erosion) of
sand in the control volume. The coordinate axis chosen in
the present situation is shown in Figure 6.13, to correspond
to other type approaches of a similar nature (125) as men
tioned previously (again note the axis change from that of
previous chapters).
The value h^ represents the entire depth over which the
profile changes and consists of the berm height above water
level "hj^^" plus the depth below water to the limits of
significant longshore sand movement "D^." An implicit
assumption in the above equation of continuity is that the
bottom profile is constant in form and moves continuously
seaward or shoreward due to the volumetric change in sand
transport caused by a differential in longshore sand trans
port. This effectively says that there is no loss of sand
to the control volume due to onshoreoffshore sand movements
but that such fluctuations in beach profiles average them
selves out to provide a net zero profile change during the
period of concern. Also any sand flux transport through the
boundaries parallel to the shore when averaged over the
236
period of concern has a net zero value. Fluxes of sand in
the onshoreoffshore direction could be taken into account
in the present model by an additional sinksource term for
gainloss to the beach profile but as no data exist;; for use
in such a formulation it will not be included. The problem
of onshoreoffshore sand transport is in need of consider
able research which takes into account both theoretical
aspects of the problejn.as weJ.1 as practical data collection
but will not be pursued further here.
Now the sand transport equation can be differentiated
with the assumption of small breaking angles as before
sina, ~ a, = 0  0, . , and an additional assumption that
b b s bi' ^
other quantities such as beach slope and wave height do not
change in the longshore direction. Therefore the continuity
equation becomes
t 8t
1
e, (sina, )" e c,
b ■ b s b
tan(j)
w
1
where
Now with
= arctan
3y
3x
e, (sina, )* e^c,
b b _^ s b
tH.
and
^b^b
tancj)
2
b .
w
3 y
3x
(6.20)
(6 .21)
(6.22)
along with the assumption that (■^)^ ^'^l we find
at ^h^
E <! ) ^
Vh^ . 2
3x
(6.23)
237
setting
(6.24)
which is the assumed constant for a given set of wave and
sediment parameters .
The equation can then be reduced to the form of the one
dimensional heat flux equation for which various solutions
exist (131 ) .
The value of K* can be referred to as a "shoreline diffusiv
ity" constant.
• The first solution discussed will be that which was
solved first by PelnardConsidere (125) under a differently
postulated sand transport model.
This situation is that of a coastal structure across
the surf zone (i.e., groin or jetty) which stops all sand
transport. For the condition of no sand transport at the
location x = 0 (the structure location) , the boundary condi
tion to be fulfilled is that = 0 , or in other words
dv ...
tanG^ = A second boundary condition xs round at x = "
where y = 0, assuming the initial condition is that y = 0
for time t <_ 0 .
The solution for this case (although with a different
value of K" ) was found by PelnardConsidere to be
(6.25)
238
y = tanG, j— exp  ( —  — ) — erfc(— — ) 1(6. 26)
^ ^/^ 2/K^ 2/K^ 2/K^
where erfcC ) is the complementar'y error function and is
tabulated in various mathmatical handbooks (132).
The solution can be made dimensionless by dividing by
the length = 2/K*t tan 6^. A solution graph for this spe
cific application is presented in Figure 6.14. The value of
K* can be found • f ro m  equation 6.2*4 and the resulting graph
used to plot solutions of the shoreline at various time
increments .
PelnardConsidere (125) also considered the case in
which the structure had filled to capacity and bypassing
began to take place. The time required for the structure to
fill "t=t^ can be found from the previous solution for the
ordinate x = 0 where Y = length of structure =
K^t, 1/2
2 (— ^) tane, .
IT D
PelnardConsidere (125) provides a second solution for
times after the structure has filled to capacity. The
boundary conditions for his second solution are that y = Y at
X = 0, and y = 0 at x = =° for all times. The initial condi
tions are as in the previous solution y = 0 at t = 0 for
X > 0. The solution to these specific boundary conditions
are as follows:
y = Y erfc( — ) (6.27)
2/K^^
239
240
which can be made dimensionless by dividing the above equa
tion by the length of structure Y. The dimensionless solu
tion is presented graphically in Figure 6.15. Pelnard
Considere used a time = t in equation 6.2 7 such that areas
of shoreline above the x axis would be equal at the time
when the structure is just filled to capacity in equation
6.25, i.e., matched solution plan areas. In this manner t^
was found to be t2 = t  G.38t^. Although the matched
solution of PelnardConsidere was not exact it appears very
useful for practical application.
Other applications of the same linearized partial
differential equation 6.2 5 can be used in prediction of
beach nourishment fill changes upon suitable evaluation of
the "shoreline diffusivity" coefficient k" . Some applications
are as follows:
— Beach nourishment fill on a straight reach of
beach. Fill exists from x <x<+x and extends Y
a a
distance from the original beach. The solution
for this specific case is as follows:
y = \ {erf ( )(1^) + erf( )(1 + (6.28)
I 2yWT a 2/K^ "^a ^
The nondimensionalized solution graph for various fill times
is shown in Figure 6.16.
— Point source type of beach nourishment such as
might exist at a location where a truck dumping of
241
■■  1 1 ■ ■ 1 1
_l
1
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i
1
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1 ■
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. ■ '
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[  
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1 ... 1 ■ 11
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"to —
to —
' — £=i
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. . 1
1

— '
i —
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242
243
sand occurs or jdrag scraping of sand from the
offshore occurs. The solution for this specific
case (where the original fill has been assumed
triangular in shape) is:
Y r
y = _£ j (lX)erf (U(lX))+Cl+X)erf(U(l+X))2Xerf CUX)+
1 , U^d+X)^^ U^(lX)^ o.(UX)
(e +e 2e
(6.29)
where X
and U
(6.30)
2/Wt
The nondimensionalized solution graph for various fill
times is shown in Figure 6.17.
For the case in which a beach nourishment has been
placed on an existing beach but a gap has been left in the
beach nourishment project (such as occurred in a beach
nourishment project on Jupiter Island, Florida in 19 74), the
following solution would apply which is similar to eq^uation
6. 28 .
y = J \evfc
— X
( (1— )
^a
+ erf c
( )(!+!)
a
(6 . 31)
The nondimensional solution graph for this specific
case is given in Figure 6.18.
— The case of a semiinfinite beach fill gives the
shape of the beach fill end and its extent of
244
245
246
247
pjTogression down tJne coa^t with time. The solu
tion for this specific case is
y = yjl + erf i— — ) (6.32)
The nondimensional solution graph is shown in Figure 6.19.
CHAPTER VII
SAND TRANSPORT AiND STORAGE AT INLETS
As the present sand transport model developed is for
ideal beaches where gradients in the longshore direction are
zero, the model would not be expected to work at inlets,
where in addition to complex bottom topography, the often
large tidal currents play an important factor in the trans
port processes. Little is known about the specifics of
sand transport at inlets (133) although generalities can
be dealt with.
It is known that many inlets act as sand traps by
storing sand in their outer shoals (133). They become self
perpetuating features by their ability to jet sand (which
comes to the inlets by ' littoral drift) to the outer shoals
where the sand is either cycled back to the shoreline during
heavy wave activity or stored permanently in the outer shoal,
making the shoal larger and consequently causing refraction
patterns at the inlet conducive to shoal growth.
In an attempt to determine what the ultimate limits of
such growth might be, a correlation was made of the sand
stored in the outer shoals with the size of the inlet as
determined by its tidal prism P and the amount of wave
activity acting on its outer shoal as determined by a wave
climatology parameter dependent on the^ energy flux at the
248
249
inlet. Details of . the sand voluiae calculatiojis are dis
cussed elsewhere (133), and the estimates of tidal prisms
and wave parameters as follows .
Tidal prism measurements came from References (134,
135, 136, 137, and 138). In most all of the cases the tidal
prism wa;s either measured from current data taken at the
throat of the inlet or by the "cubature method." Jarrett
(13 7) discusses the cubature method in detail . In most all
cases the data when the prism was measured corresponded to
the survey data from which the estimate of outer bar sand
volume was made.
Wave heights and wave periods available were average
wave heights from wave gages in the nearshore zone (15 to
20 feet below MLW) from the Coastal Engineering Research
Center wave gage program. As these wave heights already
have the measure of continental shelf slope implicit in
them (energy has been dissipated over the shelf up to the
wave gage depth) , the basic measure of wave energy used to
2 7
separate energy environments was the parameter H T~ (wave
2 2
height X wave period ). On mildly/ exposed, moderately
exposed, and highly exposed coastlines, this parameter was
arbitrarily chosen to range from 030, 30300, >300
respectively. This classification lum.ps the South Carolina,
Texas, and lower Gulf Coast of Florida inlets into the
mildly exposed coast range; the East Coast, and the Panhandle
of Florida (Gulf Coast) inlets into the moderately exposed
coast range; and the Pacific Coast inlets into the highly
250
exposed coast range. Cojrjrelations were made for three
coastal energy level groupings and for all inlets combined
using an equation:
^ = aP^ (7.1)
where
^ = volume of sand stored in the outer
bar/ shoal of the inlet (in cubic yards
of immersed sand) ,
P = tidal prism of inlet (in cubic feet),
a,b = correlation coefficients.
Linear logarithmic regression was used to obtain b for the
case of highly, moderately, and mildly exposed inlets for
the case of inlets tabulated in (9).
For these four cases the coefficient b equals:
Highly Exposed Inlets b = 1.23
Moderately Exposed Inlets b  1.08
iMildly Exposed Inlets b = 1.24
All Inlets b = 1.26
As there was no significant difference in the exponential
correlation coefficients, the value b = 1.2 3 corresponding
to the high energy coast (Highly exposed) inlets was used
for the correlations with all inlet groupings. The justifi
cation for this somewhat arbitrary fixing of parameters was
that a minimum of scatter existed in the correlation of the
Pacific Coast Inlets (Figure 7.1), The minimum scatter in
this plot over two orders of magnitude is somewhat surprising
in view of the many parameters which could be of importance
251
ure 7.1. Tidal Prism  Outer Bar Storage Relationship
for Highly Exposed Coasts
252
in inlet outer bar shoaling such as inlet history, avail
able longshore energy flux, and physiography of the inlet
coastal location.
Some reasons for some of the scatter seen are as
follows :
Inlet history. Should the inlet close
or the tidal prism be reduced drastically
(due to m.odif icat ions of the inlet inner
bay system) much of this material would be
driven back to the beaches. The author
has noted this occurrence in two occasions
on both the East Coast and Gulf Coast of
Florida. In the case of the Pacific Coast
inlets studied, all of the inlets have been
open over recorded history.
Physiography must plan an important
part also. The author has noted that inlets
i which are estuarine (i.e., estuaries), have
significantly smaller inner shoals most
likely due to the predominance of the ebb
flow during landward flooding periods . The
Pacific Coast inlets used for the highly
exposed coast correlation are alike in that
j they are estuary systems ,
253
Thus, the Pacific Coast inlets ajre .physio graphically
similar, historically open, and have similar (in a gross
sense) longshore energy flux levels; and, therefore, should
experience less scatter.
Using the exponential correlation coefficient b = 1.23,
analysis was made to determine the correlation coefficient
a and the corresponding volumeprism relationship for the
three ■ groupings of inlets and for the inlets combined.
The corresponding equations are shown below:
High exposed coasts (7 inlets)
= 8. 7 X lO'^p^^^ (7.2)
Moderately exposed coasts (18 inlets)
^ = 10 . 5 X 10"'^ (7.3)
P
Mildly exposed coasts (15 inlets)
¥ = 13. 8 X 10~^p^' (7.4)
All inlets (44 inlets)
^ = 10.7 X 10~^p^^^ (7.5)
The plots of the prismouter bar storage volume for
the various inlet groupings are given in Figures 7.1, 7.2,
7.3, and 7.4.
As inlet channel crosssectional area shows a definite
correlation with tidal prism (134, 135) and is an easier
quantity to measure than tidal prism, a correlation was also
made with the available data for bar volume — channel cross
sectional area relationships. Correlations were made for
the three coastal energy level groupings and for all
inlets combined using an equation:
254
a
rr.
<
>
1000 
500 
200 
SAN FRANCISCO 9*Y, CALIF.
MOBILE BAY, ALA . r/. V_
COLUMBIA RIVER, aRP/~ T"
NORTH EDISTO RIVER
<
03
X.
(Jj
I—
O
<5
'00 ST. AUGUSTINE INLET, FLA.f
ST. JOHN'S RIVER ENT. FLA.'
50 
GRA/'S HAR30R, WASH.
+
CA SRANOe INLET, FLA.
ST. Jf'ARY'S RIVER ENTRANCE , FLA.
GALVESTON , TEXAS
3TQ<Q0 INLET, 3. C.
NYAH SAY, S.C.
SAU SOUND , FLA.
J + PENSACQLA HARBOR, FLA.
/+
aSAU.'^ORT' INLET, N.C.
+ ^
LITTLE EGG INLET, N.J . p
COOS 3AY, ORE, ,
ARANSAS PASS, TEXAS/ iUMPQUA RIVER, ORE.
PASSAGRILLE INLET, FLA.— ^n^j. r OREGON INLET, N.C
HEREFORD INLET, N.J.— '"^/" ;+ TILLAMOOK SAY, ORE
BIG SARASOTA PASS, FLA. 4 y '^GREAT EGG HARBOR INLET, N.J.
PONCE OE LEON INLET, FLA. 4
r CAPTIVA PASS, FLA.
SARNEGAT INL^T, N.J.f
10 
DUNEDIN PASS, FLA. + 4 LONGBOAT PASS, FLA.
GASPARILLA PASS FLA...i.jy^3„,(3^;^jf^£ INLET, N.J.
NEW PASS,FLA.+^ jQ„^'3 PASS, FLA.
> ~  ^J^ r EAST PASS, FLA.
+ NEHALEM RIVER, ORE.
L>CAR0L1NA BEACH INLET, N.C.
INDIAN RIVER INLE^^ QEL.4+ CLEARWATER INLET, FLA.
2 
REDflSH PASS, rl
LU
5
_J
O
>
•0 ^ JUPITER INLET, FLA.
+/«ENIC£ INLET, FLA.
+ MIDNIGHT PASS, FLA.
0.5 
+ SAKER'S HAULOVER , FLA.
0.2 
Q I I 111 III I
12 5 10 20 50 100 300 SCO lOOO
TIDAL PRISM IN CUeiC FEET (x 10°
Figure 7.2. . Tidal Prism  Outer Bar Storage Relationship
for Inlets on Sandy Coasts
255
0.3 ' ' 111 I t I ,
12 5 10 20 50 100 200 500 1000
TIDAL PRISM IN CUSIC rEET(xlO^)
Figure 7.3. Tidal Prism  Outer Bar Storage Relationship
for Moderately Exposed Coasts
256
ure 7.4. Tidal Prism  Outer Bar Storage Relationship
for Mildly Exposed Coasts
257
¥ = a'A ^' (7.6)
c
where
= volume of sand stored in outer bar/ shoal
as before,
A = inlet channel, crosssection area at
throat (in square feet),
a',b' = correlation coefficients
Linear logarithmic regression was again used to obtain
the coefficient b' for all cases and is tabulated below:
Highly Exposed Inlets b' = 1.28
Moderately Exposed Inlets b' = 1.23
Mildly Exposed Inlets b' ~ 1.28
The value b' =1.2 8 corresponding to the highly exposed
and mildly exposed inlets was used for the correlations
with all inlet groupings, and analysis was made for the
coefficient a'. The corresponding equations for all inlet
groupings are shown below:
Highly exposed coasts (7 inlets)
V = 33.1A ^'^^ (7.7)
c
Moderately exposed coast (18 inlets)
1 9 P
V = 40 . 7A ^ (7.8)
c
Mildly exposed coasts (15 inlets)
^ = 45. 7A '^^ (7.9)
c
The plots of the cross sectional areaouter bar storage
volume for the various groupings are not shown, but prove to
have considerably less scatter than the tidal prismouter
258
bj3.r storage volume plots. Under a given set of conditions,
hence, both the volumeprism and volumecross section rela
tionships should be considered when obtaining an estimate
of the sand storage capacity of an outer inlet bar system.
A nuFiber of parameters other than tidal prism (or
crosssectional area) and wave energy also play a large
role in sand trapping on outer bar/shoals. Two important
parsmeter s which have not been explicitly considered in the
present analysis are longshore energy flux which moves the
sand to the inlet where the ebb tidal current can deposit
on the outer bar, and size distribution of littoral material
which limits the ability of the material to movement away
from the surf zone. Further research is needed to better
define how these parameters control the influence of outer
bar sand storage.
Further work is also needed on the inner bay or lagoon
shoal storage volumes and on the potential of any given
inlet to trap sand in its interior shoal system.
CHAPTER VIII
SUMMARY AND CONCLUSIONS
In the preceding chapters a model for sand transport
along beaches under the influence of wave action was devel
oped and compared to existing laboratory data and fit to
within a constant. The model predicts reasonable values for
the integrated (across the surf zone) littoral bed load and
suspended load sand transport when compared to existing field
and laboratory measured sand transport values. As of yet
though, instrumentation does not exist to perfectly differenti
ate the bed load portion of the sand transport from the sus
pended load portion (139), and thus, the individual bed load
and suspended load sand transport models cannot be verified
independently .
As both field and laboratory measurements of sand trans
port are difficult to make and very costly, few data sets
exist which provide all parameters of importance to the cal
culation of sand transport and no data have been taken in
which the distribution of sand transport across the surf zone
has been measured adequately. Also, field data taken have
often left out adequate description of the sediment size dis
tribution and rely mainly on the mean or median grain size,
whereas proper measurement should include the fall velocity
distribution curve (for suspended load) and the grain size
distribution curve (for bed load) .
259
260
Further research should be done on the following sub
jects to improve future sand transport models.
Collection of both field and laboratory measure
ments of sand transport as well as forcing functions
(wave parameters) and sediment parameters across the
surf zone.
Collection of data to define lateral "mixing"
parameter; necessary for predictive ability of dis
tribution of longshore current across surf zone.
— Concentration of suspended sediment in the
surf zone along with random fluctuations of con
centration and velocity to better define the
vertical sediment mixing parameter.
Development of better instrumentation for use
in the laboratory and field in surf zones, especially
for measuring instantaneous sediment concentrations .
Present instrumentation for use in the surf zone is
not capable of differentiating between entrained air
bubbles and sediment (139),
The present model was applied to the prediction of
sand transport along the Florida Coast in Chapter V using
existing offshore ship wave data. Computations were made
by means of computer model which takes into account wave
height modification due to energy dissipation by bottom
friction. A com.parison is made to sand transport values
estimated from dredging records at inlets which provides
reasonable agreement in most areas. By a means of data
261
presentation called the "littoral drift rose" based on
coastline orientation, the ability to postulate shoreline
stability and instability is given. Using thei "littoral
drift rose" concept,:" a rational explanation for the existence
of natural large cusp features (instability features) oc
curring along elongated bays is presented
In Chapter VI a heuristic model for the static
equilibrium (no longshop.e . sand transport) shape of shorelines
sheltered by nonerodable materials (headlands, reefs, etc.)
is presented. The model is compared to existing shoreline
shapes in the lee of reefs and shoals and is found to
provide reasonable results inrjnany areas. Using the developed
equation for sand transport, a dynamic model of shoreline
change is also postulated. The sand transport model equa
tion is coupled with a volumetric continuity (of sand) equa
tion and reduced to the "heat flux" form of partial differential
equations which is then solved for various coastal structure
(jetty, beach nourishment, etc.) cases.
Chapter VII presents the results of a regression anal
ysis for the volume of sand stored in the outer bars of
tidal inlets with the results in the form of equations which
involve both inlet tidal prism and wave activity present
offshore of the inlet. Results of the analysis allow for
equilibrium calculation of inlet outer shoal volume in the
event an inlet is to be cut in a barrier island.
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APPENDICES
APPENDIX A
DERIVATION OF LONGSHORE ENERGY FLUX PER UNIT LENGTH SHORELINE
The derivation for P,^ proceeds as follows: Assume a
coast with contours that are parallel to a straight shore
line (Figure A1) . Waves approaching this coast are assumed
to bfi. described by linear.. .smaJJ. amplitude theory. In gen
eral, a wave crest that makes an angle with the shoreline
when in deepwater will refract to make an angle a at some
shallower depth (Figure A1) , where a is related to by
Snell's law. In what follows, the subscript, o, refers to
deepwater conditions.
The path of a wave passing through point i is shown on
Figure A1 as the dashed orthogonal labeled "wave path."
The flux of energy between orthogonals in the direction of
wave travel at point i is given by
P. = C
1 ^ ^ (A1)
= (n CE)^£^,
where C is the wave group velocity, C is the wave phase
S
velocity, n = C /C, E" is the energy density, the total
average energy per unit area of sea surface,
r = CA2)
275
Orthogonal 2
Wave Crest
R3int
■ Wave Ray
u Bottom
Contours
Beach
Figure A1. Definitions for Conservation of
Energy Flux for Shoaling Wave
277
where y is the weight density of water (64.0 lbs/ft for
sea water) and H is the wave height. (This wave height is
the height of a uniform periodic wave).
Small amplitude theory is assumed, energy is assumed
constant between orthogonals in this approach and does not
spread laterally across wave orthogonals. Therefore, the
energy flux in the direction of wave travel must remain
coiistant.. beiween . ,Qrthogoxia2^,.,±ha.t . js, betweezL deepwater and
point i,
P = P. = constant, (A3)
o 1
where is defined by equation A1. However, the longshore
component of P^, designated P,^, where
P^ = P^ sin a (A14)
does change along the wave path, since a changes due to
refraction while P^ is constant by equation A3. The sub
script i in this appendix indicates any point on the wave
path, including deepwater, where small amplitude theory
holds .
From the geometry of Figure A1, it is obvious that
changes with position on the wave path. However, the dis
tance between adjacent orthogonals, b, measured parallel to
the coast does not change. Therefore, at any point, i, in
the wave path
Jl. = b cos a. (A5)
278
From equations A4 and A5, this longshore component of
energy flux can be written
P„ = (E C b cos a) sin a
(A 6)
For the straight parallel contours assumed, the distance b
is arbitrary.
Arbitrarily setting b = 1 foot (or a unit length in the
system of measurement used), one finds that the longshore
power per unit length of shoreline is :
= E C sin a cos a
or using a trigonometric identity
P„ = E C sin2a
^ 2" g
(A7)
(A 8)
At breaking, where the longshore wave energy dissipated
shoreward of breaking has been assumed to be the available
energy for transporting sand in the longshore direction
(ttC ), sin 2a,
2 g b' b
or
£b
at breaking
^Ih = 16 ^gb ^%
(A 9)
(A10)
APPENDIX B
LONGSHORE SAND TRANSPORT MODEL DATA
SUSPENDED SAND CONCENTRATION DATA
280
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Table B5. Kana Data.
SERIES STA H(ft.) T(sec. ) D(ft. ) Y(ft.) C(ppt)
10107 B02 1.15 10 1.81 .33 .141
.99 .039
1.98 .016
10229 C+l 1.48 9.5 2.38 .33 .306
.99 .097
1.98 .038
10343 PIl 1.48 8.5 2.70 .33 .323
.99 .175
1.98 .083
10564 PI9 1.80 12.0 2.86 .33 .132
.99 .081
1.98 .028
10668 CAl 1.96 11.0 3.11 .33 .259
.99 .192
1.98 .089
20109 BIl 1.96 8.0 2,94 .33 1.241
.99 .789
1.98 .351
20547 CAl 3.61 9.5 5.08 .33 1.244
.99 .702
1.98 .647
3.28 .361
20658 BU2 2.95 8.5 4.77 .33 0.499
.99 0.278
1.98 0.194
3.28 0.113
20767 PI9 2,30 7.0 3.61 .33 .258
.99 .137
1.98 .059
30216 PIl 1,80 11.0 2.70 .33 .595
.99 .301
1,98 .218
30218 PIl 1.80 11.0 2.86 .33 .376
.99 .221
1.98 .081
30442 BU2 1.97 11.5 5.20 .33 .707
.99 .280
1,98 .073
40214 PI9 1.80 9.5 4.03 .33 .149
.99 .064
1.98 .033
285
Table B5. Fairchild Data.
STA. H(ft.) T(sec) D(ft.) Y(ft. ) C(ppt)
234N 2.18 6.6 4. 70 0.20 0.328
2.18 6.6 ^.72 0,20 0.461
1.82 9.2 4. 74 0.39 0.452
1.82 9.2 4.76 0.39 0.457
2.00 8.0 4.78 0.39 0.292
2.00 8.0 4.77 0.39 0.514
1.73 6.9 4.75 0.59 0.183
1.73 6.9 4.74 0.59 0.777
1.73 ..6,9 4.72 0.78 0.105
4.72 0.78 0.320
2.00 8.7 4.61 0.98 0.463
2.00 8.7 4.59 0.98 0.185
2.00 8.7 4.52 1.18 0.235
2.00 8.7 4.48 1.18 0.351
1.92 7.2 4.30 . 1.37 0.253
1.92 7.2 4.28 0.20 1.774
1.71 6.8 4.11 0.20 1.438
2.12 6,7 3.88 0.39 1.589
2.12 6.7 3.87 0.39 0.951
325S 2.40 4.4 3.48 0.19 1.353
2.40 4,4 3.49 0.19 1.429
2.72 6.7 3.52 0.39 1.036
2.71 6.7 3.66 0.39 0.783
4.18. 5.0 4.15 0.58 1.552
4.18' 5.0 4.19 0.58 1.183
4.18 5.0 4.25 0.77 0.915
3.54 5.3 4.31 0.77 0.915
3.54 5.3 4.45 0.97 0.818
3.51 5.1 4.47 0.97 0.689
325S 3.51 5.1 4.57 1.45 0.515
3.51 5.1 4.59 1.45 0.736
4.67 1.93 0.876
4.69 1.93 0.875
2.95 5.4 4.32 2.42 0.481
2.95 5.4 4.83 2.42 0.409
2.99 5.5 4.86 2.90 0.392
2.99 5.5 4.87 2.90 0.547
3.00 5.4 4.93 0.19 3.343
3.00 5.4 4.94 0.19 3.228
286
Table B6. Continued.
STA. H{ft.) T(sec) D(ft.) Y(ft.) C(ppt)
354N 2.08 5.5 4,80 0.59 1.000
2.08 5.5 4.78 0.59 0.891+
1.86 5,4. 4.59 0.78 1.125
1,86 5.4 4.57 0.78 0.933
1.86 5.4 4,52 0.98 0.927
1.86 5.k 4.49 0.98 0.584
1.65 5.5 4.23 0.20 1.502
1.55 5.5 4.21 0.20 1.285
1.40 5,5 4.07 0.39 1.178
1.40 5.5 4.05 0.39 1.441
354N 2.20 5,8 3.91 0.59 0.681
2.20 5.8 3.86 0.59 1.568
1.65 5.8 3.75 0.78 0.703
1.65 5.8 3.74 0.78 0.425
1.65 5.8 3.62 0.20 1.101
1.64 6,0 3.60 0.20 1.714
1.49 9.,9 3.53 0.39 0.692
1.49 9.9 3.51 0.39 1.613
1,1+9 9.9 3.50 0.39 1.438
1.72 5.0 3.47 0.39 1.200
1.72 5.0 3.41 0.39 0.728
326N 1.85 9.1 2.33 0.20 0.456
1,85 9.1 2.34 0.20 2.181
1.85 ■ 9.1 2.43 0.39 1.454
1.75 8.8 2.44 0.39 0.923
1.75 8.8 2.55 0.59 0.710
1,70 9.1 2.58 0.59 0.865
1.65 9.3 2.66 0.78 0.765
1.65 9.3 2,72 0.78 0.147
1.82 9.3 2.76 0.98 0.288
1.82 9.3 2.77 0.98 0.537
326N 1.90 9.2 3.51 0.59 0.368
1.90 9.2 3.54 0.59 0.550
1,90 9.2 3.60 0.78 0.253
1.90 9.2 3.64 0.78 0.515
1.77 9.3 3.73 0.98 0.158
1.77 9.3 3.74 0.98 0.271
1.53 10.0 3.57 0.39 1.473
1.53 9.9 3,44 0,39 1.992
1.45 9.2 3.34 0.39 1.548
1.45 9.2 3.33 0.39 0.993
287
Table B5. Continued.
STA. H(ft.) T(sec) D(ft.) Y{ft.) C(ppt)
326N
1.68
9.9
3.87
1.47
0.145
1.68
9.9
3.87
■ 1.47
0.148
1.59
8.8
3.91
1.96
0.033
1,59
9.8
3.92
1.96
0.294
1.57
9.3
3.94
0.39
1.206
1.57
9.3
3.93
0.39
0.951
1.57
9.3
3.90
0.39
0.588
1.57
9.3
3.89
0.39
0.593
3i+0N 0.96 10.0 2.76 0.19 0.180
0.96 10.0 2.75 0.19 0.115
1.09 10.1 2.77 0.39 0,081
1.09 10.1 2,77 0.39 0.013
1.09 10.1 2.77 0.58 0.052
1.01 9.6 2.77 0.58 0,113
1.01 . 9.6 2.79 0.77
1.01 9.6 ■ 2.84 0.77 0.031
1.02 9.6 ' 2.85 0.97 0.085
1.02 9.5 2.87 0.97
340N 1.02 9.6 2.94 1.45
0.81 9.8 2.94 1.45
0.81 9.8 3.03 1.93
0.82 9.4 3.05 1.93
0.82 9.4 3.13 2.42
0.82' 9.4 3.14 2.42
325N
0.67
10.1
2.47
0.58
0.100
0.67
10.1
2.45
0.58
0.091
0.57
10.1
2.44
0.78
0.030
0.70
10.2
2.43
0.78
0,034
0.70
102
2.43
0.97
0.048
2.43
0.97
0.078
2.44
1.46
2.45
1.46
0.52
10.3
2,47
1.95
0.52
10.3
2.49
1.95
0.046
0.55
10,0
2.54
0.20
0.311
0.55
10.0
2.55
0.19
0.347
0.56
10,5
2.54
0.39
0.125
0.55
10.5
2.65
0.39
0.065
1.15
10.1
2.83
0,58
0.039
1.16
10.1
2.85
0.58
0.030
288
Table B6. Continued.
STA. H(ft.) T(sec) . D(ft.) Y(ft.) C(ppt)
258N l.i+U 10.3 2.54 0.19 1.274
1.44 10,3 2.55 0.19 2.380
1.20 10.1 2.53 0.39 0.283
1.20 10.1 2.54 0.39 0.498
0.74 10.4 2.71 0.58 0.165
0.74 10.4 2.71 0.58 0.158
1.43 9.9 2.77 0.77 0.076
1.43 9.9 2.77 0.77 0.174
1.07 11.8 2.76 0.97 0.078
1.07 11.8 2.76 0.97 0.112
1.30 5.3 2.74 1.45 0.094
1.30 5,3 2.73 1.45 0.102
1.30 5.3 2.70 0.19 1.712
0.99 10.1 2.69 0.19 0.751
366N . 4.30 1.45 0.083
4.26 1.93 0.099
4.26 1.93 0.047
1.87 8.2 4.24 2.42 1.073
1,87 8,2 4.22 2.42 0.087
2.12 8.0 4.21 2.90 0.148
2.12 8,0 4.21 2.90 0.097
2.06 8.5 4.20 0.19 1.474
2.06 8.5 4.20 0.19 1.477
1.92 8.1 4.21 0.39 0.207
1.92 ■ 8.1 4.22 0.39 0.015
2,12 7.5 4.23 0.58
2.12 7.5 4.26 0.58 0.040
APPENDIX C
ANALYSIS OF SSMO WAVE HEIGHT, PERIOD, AND DIRECTION RANGES
The purpose of this appendix is to describe the manner
in which the groupings of wave data listed in the SSMO
volumes were handled for computations of longshore energy
flux.
Wave Height
For the SSMO data, a representative value of H^ must
be chosen for each interval of wave heights contained in
SSMO Tables 18 and 19. Since energy is a function of wave
height squared (in linear theory), a representative value of
H^ for a given range of H^ values should be based on the
mean square root value of the wave height over the range.
Consider the probability of occurrence of a wave with
specific height H as equal to p(H) in the range H^ to H2.
The energy represented in this band of wave heights is
proportional to H;^ the mean value of a representaxiive wave
height squared where :
J p(H) H^ dH
h' = ^
r fH2
p(H) dH
^1
289
290
Since pCH) is not known,, it J_s .cnnsi dered uniform,
which is reasonable if the wave height range < H < H2
is small. The equation then becomes:
H^dH
1(H2  hJ)
3(H2  H^)
(II2)
dH
Taking the square root of this value,
1/2
H
1(H2  hJ)
_3(H2  H^)_
(II3)
Using Equation (II3), representative values of were
found for the corresponding ranges of given in SSMO data,
and are summarized in Table C1.
Wave Period
Representative values of T were assumed to be the
average of the SSMO period ranges, and are given in Table
C2. For T > 13.5 seconds a representative value of T = 16
seconds was assumed.
Wave Direction
Directional observations as recorded in the SSMO volumes
are given on eight points of the compass and thus correspond
to eight M5° sectors of the compass. In the computation of
the longshore energy flux, the midpoint of the sectors, as
291
Table C1. Representative Values of Wave Height Used in Computation of
Longshore Energy Flux.
Actual SSMO Range of Heights Height Used in
SSMO Coded Height (feet) Computation (feet)
<1
12
.82 
0.82
2.46
0.47
1.71
31+
2.46
4.10
3.31
55
4.10
5.74
4.94
7
5.74
7.38
5.58
89
7.38
9.04
8.22
1011
9.04
10.70
9.85
12
10.70
12.30
11.49
1316
12.30
15.50
13.98
1719
15.60
18.90
17.25
2022
18.90
22.15
20.53
2325
22.15
22.43
23.81
Table C2. Representative Values of Wave Period Used in Computation of
Longshore Energy Flux.
SSMO Coded Period Used in Computation
Period
Actual
SSMO Range of Periods
of Longshore '
<5
0
< T < 5,5
3.0
67
5.5
7.5
6.5
89
7.5
9.5
8.5
1011
9.5
11.5
10.5
1213
11.5
13.5
12.5
>13
13.5
CO
16.
292
givexi in the SSMO data .by ..tlie. eight pn±Qts of the compass,
were used as the representative values of 6 for direction of
wave approach. When a representative wave having a given
frequency was parallel to the coastline, the corresponding
sector of waves was divided into two parts , one being de
leted from the computation and the other approaching the
coastline from the midpoint of its half sector with the
coTTresponding frequency halved (see Figure C26). Wave data
in octants with midpoints in the offshore direction
(> e + SQ°) for a given coastline orientation have been
n —
deleted from the drift computations.
In the SSMO data it was ascertained that a considerable
number of the original observations were taken on the 3 6
points of the compass, and, when reduced to the eight points
of the compass in the SSMO tables, a skew of the wave direc
tion was introduced. This skew amounts to a ten degree
shift clockwise, and has been accounted for in the results
of the littoral drift computations.
293
Wove data offer modification
Figure C26. Modification of Wave Data for
Waves Parallel to Coastline
Average Annual Total Littoral Drift
Diagrams Along the Florida Peninsula
295
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cn
BIOGRAPHICAL SKETCH
Todd L. Walton Jr. was born on March 17, 194 7, in
Dayton, Ohio. He graduated from Belmont High School, Dayton,
Ohio, and then entered the University of Cincinnati, in
Cincinnati, Ohio. While in this cooperative college, he
worked as an engineer with the Baltimore and Ohio . Railroad
and the Southwestern Portland Cement Company to pay his way
through school. He graduated in June 19 70 with a Bachelor
of Science degree (with honors) in Civil Engineering. After
undergraduate school he enrolled in a graduate program at
the University of Florida and completed a Master's degree
in coastal and oceanographic engineering (with high honors)
in June 1972. During the years from 1970 to 1972 he also
completed a Reserve Officers Training Course.%and was com
missioned as a 2nd Lieutenant in the U.S.A.R. Upon gradua
tion, he fulfilled his military obligation with the U.S.
Army Engineers at Ft. Belvoir, Virginia, and joined the
faculty at the University of Florida as an assistant
engineer.. In the years since graduation he has taught
classes in coastal structures and littoral processes, worked
in research in the area of coastal hydrodynamics and beach
erosion problems, and served as the Coastal Engineering
Specialist to the Florida Sea Grant Program, as well as
321
322
serving on a Governor's Task Force (Florida) on Beach
Erosion and Inlet Maintenance. He then decided to pursue
more graduate studies.
"For thou hadst cast me into the deep, in the midst
of the seas; and the floods compassed me about; All thy
billows and thy waves passed over me" (Holy Bible.
Jonah II : 3) .
I certify that I have read this study and that in niy opinion it
conforms to acceptable standarxis of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
B. A. Christensen, Chairman
Professor of Civil Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
T. Y. Chiu>^
Associate Professor of Coastal and
Oceanographic Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
J. G. Melville
Assistant Professor of Civil Engineering
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Daniel P. Spangler
Associate Professor of Geology
This dissertation was submitted to the Graduate Faculty of the College of
Engineering and to the Graduate Council, and was accepted as partial ful
fillment of the requirements for the degree of Doctor of Philosophy.
June 1979
Dean, College of Engineering
Dean, Graduate School